By itself, it would only help a little. The key next time is to win back third party support. Power does matter for that, but it's not the only thing. They also have to improve their online infastructure, do away with gimmicks like the gamepad (I love it at least but I realize it's not a hot selling point) that make it harder for developers, etc. And they need an actual HDD with lots of space given that we are in the era of 10 GB plus updates.

Funny, that's true, they are the only Nintendo home consoles I never owned.  But that was for other reasons, I didn't buy N64 because I was busy with University and I didn't buy GC because it was a little purple box that played games called Mario Sunshine and Toon Zelda on child-sized discs.  This new beast machine will not be purple nor have a handle.

GC was a disaster, but N64 was not.  It was fairly competitive with PS1 in US sales and nearly matched its predecessor the SNES in that region.  It failed to be the market leader because of the lack of CD format for media and subsequent loss of Final Fantasy and other key 3rd party titles which had a major effect on Japanese support.  It also launched later than PS1, just like GC launched later than PS2.

This new system would be launching very early, not unlike the PS1, PS2 or XB360 did. It would have a very cool, mature design.  It would be built to meet the needs of third parties well into the future and be easy to develop for.

I believe it wouldn't make any difference.
It could pull some of the hardcore people, but won't actually put down impressive numbers.

Nintendo's problem is deeper than the hardware

It would depend on how good the third party support was. I'm not to particular found of many nintendo games other then the classic Snes donkey kong games and Zelda.