Now, this is an interesting topic.
I'll start with the comparison of consoles' graphics. Now, when comparing the Gamecube's graphical capabilities to any modern-day visuals, namely to it's predecesor the Wii, there is a bigger difference than you might think. First of all, make sure to compare games that arrived in a similar lifetime as on the other console; comparing something such as Super Mario Galaxy, which arrived quite early in the Wii's lifetime, to something such as Zelda's Twilight Princess (since it can be run on a 'Cube, I consider it a 'Cube game as such) is rather detrimental. Rather, comparing Super Mario Galaxy to a Gamecube game's graphics from 2002, since both games would of been released a year after their home console was.
So, let's compare Super Mario Sunshine to Super Mario Galaxy. I have personally played both, so I can easily support Mario Galaxy's incredibly superior graphics. Youtube isn't the most ideal for quality comparison, but let videos buffer for a time in the highest quality for these videos.
Super Mario Sunshine:
Super Mario Galaxy:
I couldn't even find a good quality Mario Galaxy video, but I'd say it's rather clear that Galaxy's graphics are cleaner and support far more polygons than Sunshine's, even when you put it into 720p. I'd go far enough to say Sunshine's "cutscenes" are of less quality than Galaxy's typical gameplay graphics. Even the latest released Gamecube games would still appear lackluster to Mario Galaxy, if you ask me, and the Wii's graphics have improved hugely since then, such as with Metroid: Other M. The cutscenes in this preview are what the Wii does indeed show, except better.
The graphical jump from Xbox to 360, and PS2 to PS3 is also quite major, especially with the PS3 utilizing BluRay towards better graphics period. While graphics are anything but the most important aspect of a game, it certainly should be clear after those examples that this generation has massively improved a system's visual capabilities.
Onto the more important part of this perhaps soon-to-be debate, the gaming aspect of the new generation. The technology behind our new systems does more than support updated graphics. If you tried running Call of Duty: Black Ops on a PS2, you'd get nothing more than a crashed console, sort of like trying to run Vista on a 1995 computer and OS; it just doesn't work.
As the engines behind our consoles improve, so do the possibilities with all their other features, notably as you've mentioned, online play. I've always found the internet surfing and being a psuedo-media player as a rather unnecesarry but still interesting feature of our new consoles (yes, even the Wii if you've the mind to hack it via homebrew). But the actual online gameplay is where the importance of a console's new capabilities lie.
Not counting how taxing a game itself is on the system's engine, but what if you tried to run the most infamous of all multiplayer shooters on a PS2; MAG? This game can support up to 256 actual players in a single online game, working in real time via headset, on-screen HUD orders issued by your human commander, and so on. Even with incredibly dumbed down graphics, the PS2 simply wouldn't be able to handle all that activity, no matter how great a server you give it for the game.
If last gen's systems did remain in development for long enough, I'm sure online play could have been much more explored and impressive when being compared with today's standards, but that's simply not the case. Thus, developers needed room to think and inovate, and so came our new consoles.
I hope this at least helps support my position that our new generation's consoles are incredibly more powerful than the last. I can ironically agree on the release of a new generation being detrimental in one way, though; being released too early.
The major gaming companies themselves claim to support the 10 year cycle for a system, but we havn't seen a console with such a life span since the N64. The fastest developing countries are the most avid gamer fanbase, and are basically over run with the urge to have powerful technology, and have it updated constantly. The world constantly sees new "improvements" and implements in cell phones, computers, programing and ofcourse, video games. It's a shame the video game industry has been consumed by the demand for constant, more powerful technology, since if the trend continues, we'll never see the true potential of our consoles.
We also won't be able to see the best starting points of consoles. The 360 was terrible because of it's RRoD when first released, we all know this to be true. The PS3 had next to no games apon release, and none of them were epic launch titles like Nintendo almost always supplied. If I recall, none of them were even Sony's A-team 1st party games, although I may be wrong.
Given a few extra years, the PS3 would of clearly had a bigger game library, and the 360 might not of been so fail at applying a more powerful glue. So while I certainly don't say the leap to generation 7 was unnecesarry, I will claim it was too early.
EDIT: Ahh, the good ol' wall of text. Great to see it again.