As someone who was already 24 when the N64 came out, I recall three feels:
1. Mario 64 was amazing, WTF was this three-dee graphics omg.
2. Cartridges were kinda cool in that they already seemed retro but... games sure are expensive...
3. Sure is taking a long time for software to come out... and Playstation is getting all the cool stuff... and Square...
N64 was the end of the party really. It kinda blows my mind that so many people complain about the wii remote, or the Wii U pad, while hazy memories of the N64's batshit insane pitchfork pad was just the dumbest design ever, analog stick or not. For all no loading time is great, N64's vasoline graphics and fog got old quick, and its software library was depressing. The "hardcore" game world quickly steered around it and left it behind.
The Gamecube, by comparison, was amazing and not just because it had better graphics. It's easy to see how it could over-represent itself in people's memory, especially if you were younger when it launched. But the Gamecube era was marred by frustration, as well. It got a lot of 3rd party software, true, including a few leading games. But there was always the feeling that 3rd parties didn't care. Ports were usually mediocre in some way, no-effort. Looking back people like to point out Gamecube was superior to Wii because it got multiplatform games - but they don't mention that it often got the worst versions.
And most of the really desirable games passed it by. Even the Xbox got ports of games like GTA3 and San Andreas, if delayed.
There's a reason why I like the Wii so much. For all that Nintendo made a few errors IMO, such as lowballing the hardware just a tad (could have used 720p output for Dolphin-like graphics), Wii was actually a cohesive vision. Rather than trying and failing to compete with everyone else, it was a fully encapsulated Nintendo experience. Their own thing from top to bottom.
While it was missing a few star games from the GC, it more than made up for it with superior 3D Mario and Zelda games, a great Fire Emblem (so often forgotten), even multiple Kirby platformers and a Donkey Kong game. It got a lot of very nice niche Japanese games. Its two biggest problems were: not launching with motion+ to allow better games to be made from the start, and Nintendo getting their usual cold feet towards the end of a platform's lifespan. If Xenoblade, Last Story, and even Pandora's Tower had been released normally worldwide, perception of the Wii may have been a bit different for fans. Instead of nothing but frustration.