The budgets for those games came at a cost during the Gamecube generation, which would have eroded profit margins at the time.

The WiiU added another 2.4~ million copies to Wind Wakers sales and now has rolled into Nintendo's subscription service for recurring revenue... So some of the expenses during the Gamecubes life is being recouped even today.

Marketshare isn't everything, profits are... Because that ultimately dictates support, re-releases, remasters, remakes and sequels in the end.
Microsoft is making more money than Nintendo or Sony and they sell the least amount of hardware... And because of that, they can still justify another console generation.

Gonna disagree there, a platform is a success or not in and of itself, a game from it being remade like 20 years later isn't part of that console. 

Gamecube was outsold 8:1 by its chief competitor, saw Nintendo's marketshare at its lowest point up to that date, and I haven't seen any data to indicate it was profitable.

We're kinda going way off topic here though, as this is a thread about Xbox.

Correct. We haven't seen data that determined if the Gamecube was profitable or not. That wasn't my argument.

My argument (And this applies to ALL platforms) and why the OG Xbox is regarded as a failure is due to profitability.

If a company doesn't make money from a product, then the product is a failure, doesn't matter if it sells 1 million units or 100 million units.

Now we can quantify "success or failure" in other aspects, like the Original Xbox was a success in enabling Microsoft to enter the console market, but it was still a  failure, they lost billions... And companies ultimately exist to make money.

All gens are unwinnable for Xbox. If they couldn't pull it across the finish line with the 360 (just barely losing to PS3 at the very end), they can never do it. They simply occupy too much of the same territory Sony does + Japan and much of non-UK Europe doesn't care for MS's hardware for whatever reason. I'm an Xbox guy, these days even more than a Nintendo guy, so this isn't coming from trolling or provocation but just saying it like it is. Wish it wasn't the case, but that's the way it be. Especially lately, there's little to differentiate them from PS consoles or PC. Nintendo has been a mixed bag of late but they've done are great job of finding their niches and leaning more on those.

That's not to say MS can't come up with a sustainable model - personally I think they'd do well continuing to focus on the strength of their GP service, online, and utilizing multiple smart devices/hardware to play their content going forward, where Sony seems more focused on the PS consoles on the TV/home theater more heavily as *the* way to play PlayStation games.

Xbox I would say is arguable as while it did lose money, as you say it did also manage to open up a new market to MS. At the same time though, it did get crushed by the elephant in the room.

On the flipside, even if Xbox Series turns a profit, it could be argued as having failed as it has effectively killed Xbox's console business.

I'm surprised so many many people have a defeatist attitude here.
The reason Xbox cannot win, is because Sony is around? How is that a valid statement.
Isn't that just, normal competition?
Surly the American would say, that's capitalism, just do better than your competitor!
It sounds odd to hear people here go "yeah they had no chance".
That is like saying they cannot ever better themselves to be equal or better than Sony.... why?

It hasn't killed the Xbox business though. Microsoft has made $80+ billion dollars since the Xbox series released.

It's sales are higher than other highly rated machines like the Atari 2600, Nintendo 64, Sega Genesis, Original Xbox, Gamecube, Dreamcast and more... And likely just to come up a bit short of the Super Nintendo.

...And as such, is enough for Microsoft to justify another Xbox generation... And also justify partnering with Asus and other OEM's to build a handheld ecosystem.

Is it going to outsell the Xbox One or Xbox 360? Not a chance in hell. But while it's profitable (And Microsoft is not chasing a $4 trillion market cap for nothing) Microsoft will keep pushing it.

Remember, companies exist to make a profit.

With the way Xbox Series sales have slowed to near zero, there's a real risk that next generation could be DOA.

Hopefully not, but there's an argument to be made that the Series is a failure despite being profitable, in so much as their console business may not recover.

Everyone knows that those $80~ billion (revenue, not profit) aren't attributable to the Series consoles. They're mostly the result of acquisitions and transformation to a gigantic 3rd party publisher. It cost them over $80 billion to get there, meaning their net gaming profitability will be in the minus for a very long time. All MS has is its sheer ability to brute force, which is gained from their dominance outside the gaming industry and has absolutely nothing to do with their consoles.

Sony gaming's revenue (popularity metric) even after MS gobbled up half the market (hyperbole intended) remains comfortably higher. And Nintendo's profit margins (success/efficiency metric) are on a different level. We have no clue how much profit (even when acquisition expenses are excluded) MS is making from gaming, and even if we did, the Series consoles are factually not responsible. MS gaming is not restricted by the failure that is Series X/S.

I never made the assertion that was profit from the Xbox Series consoles alone. Hence why I said "Gaming Division" and not "Console Sales".