The net result of the slow processor was they gave Sega a foothold in the market while weakening their own brand and destroying what had been a monopoly. The net result of carts was handing the market over to Sony, something which I'm not sure can ever be erased. We'll see if it works this time, but Nintendo's main concern isn't either sales or gameplay so much as profitability. Not a bad thing, but it's not likely the way to the top either.
This is all wrong. The supposedly-outdated-in-1983 NES was easily trouncing the Genesis before the SNES launched. When Nintendo seperated their focus between the NES and SNES was when Sega gained a foothold. Nintendo then botched the whole generation by fighting the war on Sega's terms. IE, focusing on "power" and fighting over young teens as the main market. The irony of the SNES/Genesis generation being "the first great console war," is that both companies really flubbed up the whole thing, compared to what they could have done.
With N64, it was Nintendo's policies towards third parties that killed them. If you don't know about those, then read up. They were basically trying to control the growth of all third party developers and publishers with a series of illegal monopolizing policies. It was no surprise third parties raced out the door when Sony provided a real alternative (ie, an alternative that had differentiated themselves from Nintendo... Sega hadn't.) The expense of carts cutting into potential profit margins certainly accelerated the exodus, however.
Furthermore, Nintendo's main concern is none of the things you mentioned, but rather it is reaching new markets. This isn't PR speak, its literally the new vision and focus that Iwata has brought to the company. Iwata had a great quote something along the lines of "even if we sell the most, we must always fight consumer apathy towards video games."
Agreed that Gamecube was the first attempt by Nintendo to compete technologically, and I think that's why we're seeing what we're seeing with Wii. It's not the PR spin that they give us. It's that nobody cares whether they have the technology or not, so they might as well not. That said, the improved technology with Gamecube did reverse the mass exodus of developers from the Nintendo camp... to an extent.
The development community only came back to Nintendo with Gamecube in that it was a generation of multi-platform titles. The system still only had a tiny fraction of PS2's support. GC was by far the best hardware last gen (both cheap and powerful), but it didn't matter in the long run.
Nintendo was still popular with the public and quickly passed Sony's sales in the US during the year after its launch (coming very close in total sales). However, developers had their minds set regardless of this, and they weren't budging just on the basis of a few strong months. The public eventually followed, after realizing that they weren't going to be able to get anything but Nintendo games on a Nintendo system.
But right now, you ARE seeing developers budge. Every third party developer and publisher now sees the Wii as an integral part of their long term strategy--this is a stark contrast to GC. The support is obviously not up to PS2 levels yet, where developers' whole livlihood relied on PS2.