As a hardware developer, you should always try to make it a point to appeal to 3rd party developers regardless of what you want to do with your system. Because there are going to be stretches, especially when you're making that next gen leap, something Nintendo has struggled with at times (N64 -> GC; DS -> 3DS; Wii -> Wii U), where your own 1st party software is not going to be there. The release schedule will dry up a bit or worse, you rush the development of your games to make sure the release schedule isn't dry and the quality of those games suffer as a result. As such, you need to make sure your hardware is appealing and attractive to 3rd party developers as well as doing what you can to support and market their games on your system so that they can take the pressure off you and carry the torch for you for a bit until you've got your feet under you and are ready to take the torch back and lead the sprint yourself!

The biggest mistake Nintendo (Yamauchi) ever made, even more than the Wii U's flop, was going with cartridges with the N64 instead of CDs, alienating 3rd parties to the point of driving them in mass towards the PlayStation - Square, in particular, as Final Fantasy VII - which was originally supposed to be an N64 game, got the opportunity to really shine and prosper on the PS1 and with Sony's big bucks and marketing behind them that they NEVER would have had if they stuck with Nintendo. And that's the direct consequence of them showing little regard or consideration for 3rd parties on the NES and ESPECIALLY the SNES. They prioritized themselves and pushed their "My Way or the Highway" mentality on them for those two systems. As OP stated, it was necessary earlier on following the crash of 83. But by the time of the early-mid 90s, when SEGA and especially Sony entered the fray and provided viable alternatives. Nintendo's practices, little regard, and sometimes outright dickish behavior towards 3rd parties was just completely unnecessary, and it finally bit them in the ass when they left in droves during the N64's life span and that system suffered as a result. A mistake they tried to rectify with the GameCube, but at that point, the damage was done and Sony firmly had 3rd parties in their grasp and Microsoft had stepped in as a viable option as well.

So, while I can see what Miyamoto is saying in regard to his regrets with the GameCube, I don't entirely agree with it. At that point, Nintendo had finally realized that in order for their system to be viable, they needed to have strong 3rd party support, and they couldn't just rely solely on themselves. Them trying to get back in 3rd parties' good graces with the GameCube was absolutely the right decision. The problem was that the damage was done at that point. The writing was on the wall and Sony was the darling hardware developer now with Microsoft being a favorable partner too. And no matter how hard Nintendo tried, they were not going to win those developers back from Sony and Microsoft. Their more limited funds compared to those two giants being a key reason like OP stated - Nintendo didn't and STILL doesn't have the room and cushion to sell high-end premium hardware at a loss like Sony and Microsoft do. But even if they did, 3rd parties would have still leaned with Sony and Microsoft just because their games had found a solid audience and support on those platforms that they were never going to get on Nintendo. Because at the end of the day - Then, Now, and Forever, Nintendo's #1 priority was, is, and always will be... Nintendo! If Final Fantasty VII, and subsequently, VIII and IX, had stayed on the N64, however those games turned out and even if they still went on to be the great games they were, they would NEVER have reached the heights it did on the PS1 because Nintendo was NEVER going to push those game over Super Mario 64, Mario Kart 64, GoldenEye, Ocarina of Time, Majora's Mask, Donkey Kong 64, Diddy Kong Racing, Banjo Kazooie, Super Smash Bros, etc. No matter what, Nintendo will always put themselves first. And EVERY 3rd party publisher knows that and that's why even now, they prioritize Sony and Microsoft because they know they'll get far more support and backing from them than they will from Nintendo.

And early on in the GameCube's life cycle, Nintendo realized that was the case and that they were fighting a losing battle trying to beat Sony and Microsoft at their own game. It wasn't going to happen. So that was when Nintendo had to make a change in their strategy and pivot more towards trying to be innovative, revolutionary, and make their hardware stand out from the competition. While also developing their games in unique ways catered towards that hardware to really make them stand out and still look attractive. And that's what led to the DS and Wii.

And... it worked! Before the Switch, that Wii/DS era was hands down the most successful period in Nintendo's history. Because instead of trying to chase trends, fight a losing battle, maybe sell more expensive hardware at a loss, and be less like themselves to try to court 3rd parties who weren't all that interested. They doubled down! They made themselves even MORE unique and out-of-the-box and managed to WOW the general audience. To the point where those 3rd parties, even though they were doing just fine for themselves on PlayStation and Xbox... They couldn't just ignore what Nintendo was doing with their systems, because there was an audience and opportunities there too! And to just ignore them would have been to leave a lot of money on the table! So Nintendo, seemingly against all odds, found a way to have their cake and eat it too! They were able to get 3rd parties to come back and support their system, even if it's not quite on the level of their support for Sony and Microsoft, it's still there and it's that much more money for them to make! While also making sure their games are still the top priority and main attractions and reasons why people buy their systems, AND make sure that they can still sell those systems at a profit so now they make more money than they ever have before. It's a win-win for everybody!

However, the problem with beating to your own drum is that it is a double-edged sword. When it works, it REALLY works and you reap ALL the rewards and benefits. As was the case with the DS and Wii.

And when it DOESN'T work... It comes back to bite you in the ass. Which is exactly what happened to Nintendo with the Wii U and to a much lesser degree, the 3DS. 

For the first time in their history as a game developer, maybe ever (to my knowledge) Nintendo was LOSING money. Posting operating losses. All because they made the same gamble on the Wii U as they did the Wii, only this time it backfired tremendously. All because they misread the market and made the mistake of leaning too much on the casual market that carried the Wii, and to much lesser extent, the DS, in the 7th gen. A market that had since then migrated to the mobile market and weren't all that interested in dedicated gaming. So the audience that was so captivated and enthralled by the Wii and DS... didn't really care all that much or just were confused (that's another topic for another day) by the Wii U and 3DS, and the drastic drops (100m to 13m and 154m to 75m) reflect that.

To make matters worse, Nintendo had severely underestimated just how demanding HD development is and so they ran into the very issue I mentioned earlier about not being able to maintain their release schedule, leading to some very dry and barren lineups... An issue that 3rd party developers could easily help with... The problem with THAT is, they're only as interested as in your gimmicky system for as much as the system itself is profitable and lucrative... Which is to say the Wii U was... Not at all. And neither were these developers once they saw the writing on the wall. So Nintendo was on their own with a flop of a system they had no hope of redeeming. 

HOWEVER, it is in that failure where their decision to double down and bet on themselves was starting to pay dividends. Despite the Wii U's overall hardware and software sales being less than that of the GameCube, the attach rate, especially for Nintendo's own first party games, were noticeably stronger and eye-catching. 13 million consoles sold, but over 8 million copies of Mario Kart 8, an attach rate of around 60%. And the 3DS, despite selling less hardware, and if I had to bet, less 3rd party support, than the Game Boy Advance, still managed to sell more software, thanks in large part to Nintendo's own offerings carrying the load.

So even though they had fallen flat on their face and were in the midst of dusting themselves off, Nintendo still had THAT to hang their hat on and be proud of. The fact that they had managed to build their loyal base and following that will invest and ride with them as long as the quality of their games are still top-notch. Which is a direct result of them prioritizing themselves over 3rd parties. 

That just left the matter of figuring out where they went wrong with the Wii U and trying to make sure they right the ship with their next system.

So Nintendo, trying to cleanse themselves of the stench that was the Wii U's failure. Had to figure out where they succeeded with the Wii/DS while also figuring out where they went wrong with the Wii U/3DS. While also trying to repeat what they managed to each back in the mid-2000s of standing out in a way that made them extremely attractive, lucrative, and desired when the Wii/DS hit the market. After many considerations, evaluations, and planning, the end result was what will now go down as the best-selling Nintendo system of all-time, and what, at the time of this post, has a very real chance at becoming the greatest selling system EVER - The Nintendo Switch.

And they did it by, once again, doubling down and betting on themselves and their way of doing things, while making sure not to repeat the mistakes they made with the Wii U/3DS. And now as we are on the verge of the reveal and eventual release of Switch 2. Nintendo's prospects look brighter than they have ever been! And that can be seen, not only in their massive hardware sales for the Switch, but even MORESO in their software sales on the system, ESPECIALLY their own software. With a Top 10 list where 9 of the 10 games have sold over 20 million copies each and the entire Top 20 have sold over 12 million, ALL 1st party games - Nintendo's digging their heels in and putting themselves first has never paid off more gloriously for them than it has with this system. And it's not just them, but 3rd parties, like they did with the Wii/DS have come around and are wanting to take their own slice of that Switch pie. Only THIS time, it's on a much more dedicated, concentrated level than the Wii/DS. We have seen 3rd parties, more and more, try to make Switch versions for their major releases alongside their usual PlayStation and Xbox versions. And with the Switch proving to be a huge success and draw amongst dedicated gamers as much, if not more, than casual gamers, that investment is paying off for them as well. Stardew Valley and Monster Hunter Rise most noticeably being among the best-selling 3rd party games on a Nintendo system.

Yet again, Nintendo has found a way to have their cake and eat it too. Only THIS time, it looks like that cake won't be running out for the foreseeable future.