I disagree with both. PS2 had more third-party games than Xbox or GameCube, but there was still some overlap. PS2 first-party games could be a deciding factor, even if only for 5-10 million customers.

I think N64, GCN, and Wii U sales show there is a floor for people buying Nintendo platforms almost exclusively for Nintendo games. And I think that floor for Nintendo home consoles is around 5-10 million customers. 

In all fairness, I think across the industry there are about 5-10 million customers in that broad range who will buy a platform for exclusives. Saturn and Dreamcast both sold over 5 million but below 10 million. 

I wasn't a fan of most of Sony's first party games until the PS3. the PS1/2 was all about Square and Capcom for me. The PS3 was where Sony really hooked me with Uncharted games Last of Us, and many others.

Yes. Tho I understand that the impression we have on that console is that the 3rd party support was its main strength (cause it was massive) the first party games it got were also goated and it was a deciding factor for me at least to get the console.

That 5 to 10 million is important though because they are the 1st adopters and the people who tend to buy a lot of games. Losing a few of them is like losing ten regular customers. They also act as free advertising via word of mouth. This is why an early launch lineup of exclusives is so important. Switch had it and sold 150 million units. PS5 and Xbox didn't really have it and so sales of those systems have stayed the same as last gen or been on a huge decline. 

Wii U had a higher software attach rate than N64, despite having fewer physical games. Shoot, Wii U has a higher attach rate than Dreamcast, 3DS, etc.

I'm sure Nintendo would rather have Wii U sales end up like they did than 18-24 million pieces of hardware sold with greater losses and less profit.

5-10 million enthusiastic customers (no matter how casual or core they are) are indeed a lifeblood to a platform that it needs in addition to tens of millions of less enthusiastic customers. 

Despite the fact that it won't have exclusives, at least not console or true exclusives, I do think something like 5 million or, so units are the floor for Xbox 5. While a small group in the world, enough gamers and Xbox fans would show up to hit that amount. Shoot, even something like Steam Deck that is only sold by Valve firsthand in most markets has sold over 4 million units and is probably over 7 million units by now. It is a handheld PC, but it wants some vibes similar to consoles. 

In terms of total volume of games, technically it was the PS2, but a large portion of those games were the large volume of titles put out by 989 Sports and London Studio (the latter of which made a ton of EyeToy games). Relative to the size of the total library, then definitely the GameCube, since the PS2 had an insane number of games on it, perhaps more than any other console (if we exclude small digital-only games on more recent systems).

But when it comes to games large numbers of people actually played, the GameCube probably had somewhat better first-party support. Which makes sense, considering Nintendo is far more reliant on their first-party output than PlayStation is on Sony's output. In fact, the differences in the first-party situation were even more stark back in Gen 6. While the list of best-selling GameCube games is dominated by Nintendo games, that same list on the PS2 is primarily third-party games.

Aside from Gran Turismo, there weren't much in the way of big marquee first-party titles on PS2. Aside from GT, there was also God of War, Jak & Daxter, Ratchet & Clank, and SOCOM. While GT3 sold 14.89M copies and GT4 sold 11.45M, the next bestselling games were God of War at 4.62M, God of War II at 4.23M, and Jak & Daxter: The Precursor Legacy at 4.2M. Everything else outside the Top 5 first-party PS2 games (not counting the game that was bundled with the Eye Play accessory) sold under 4M copies, with most at under 3M. All told, there were about 20 first-party games on the PS2 that sold at least a million copies. This is on a system with a final install base of 160M. The original Gran Turismo was the only first-party PS game with a final attach rate of over 10% until GT5 and Uncharted 3 joined the club. So, while It wasn't until the PS3 and especially the PS4 that Sony saw their first-party output outside of Gran Turismo really start to explode in terms of popularity.

The GameCube, meanwhile, had 27 games that sold at least a million copies, all of which were first-party, and of which eleven were multi-million selling titles. This is despite having a final install base of only 21.74M units, less than one-seventh that of the PS2. It had nine games with an attach rate exceeding 10%, and of those four of them had an attach rate of over 20%. Melee sold 7.41M copies, giving it an attach rate of over 34% and making it the fourth best-selling first-party title of that generation behind only GT3, GT4, and Halo 2. Every since the N64, first-party games have been the main event on Nintendo systems, and to this day they routinely have first-party games with sales volumes and attach rates that few non-Nintendo games have ever achieved on any platform. I've often said that people get Nintendo systems mainly to play Nintendo games, and the numbers back that up.

TL;DR: It depends on what metrics you want to use.

I agree that PlayStation generally isn't as reliant on First party the same way Nintendo is. That said, first party has still always been important to the brand as much as third party has been.

However, the point I'm arguing is that from the perspective of a gamer in the early 2000s, you see Nintendo constantly disapointing and under delivering with most of their major GameCube releases game after game, and having spread out way too far in release. Meanwhile, you see Sony pumping out genuine classics, or at the very least, decent games, on a much more regular and consistent basis for PS2. The later begins to look like the better option in comparison at the time.

Shadow of the Colossus and God of War 2 are some of the greatest games ever IMO.
I did not play Gran Turismo 4 but always see people still playing it to this day.

Wind Waker looks great, and like GT4 Super Smash Bros. Melee is still played to this day because of tournaments.
Aside from that I think there is a good bunch of interesting Nintendo games maybe more overall individual games than Sony had, but none that I think would reach that high.

I have basically no experience with Gamecube games tho. But even so, I would pick the PS2 just for the exclusives I know from both.

GC beats out PS2 in that department quite comfortably imo it's easy to forget the first party output on GC, off the top of my head:

Mario Sunshine
Metroid Prime
Metroid Prime Echoes
Animal Crossing
Wind Waker
Twilight Princess
Luigi's Mansion
Eternal Darkness
Paper Mario TTYD
Fire Emblem Path of Radiance
Smash Bros. Melee
Chibi Robo
Pikmin
Pikmin 2
1080 Avalanche
Custom Robo
F Zero GX

PS2 had Sony's best first party output of any console they made. A lot of great stuff and variety. I feel Nintendo's output on Gamecube was up and down. Sunshine the worst 3D Mario. Double Dash the worst 3D Mario Kart. TP is a bland Zelda game. Wind Waker was amazing. Prime was amazing. Smash Melee was amazing. When Nintendo hit that gen like any gen they made all time classics. I don't think even the best PS2 first part games hit Nintendo's highs. Sony had some bland games like Primal. So I guess it just depends if you like a lot more variety with some very good games and some so so games or peaks and valleys of Gamecube but those peaks soar above the clouds.