Yeah I'm a semi-hardcore gamer, and I'll only have the Wii this gen. Well that and a PC. And I've never owned a console before in my life (though my brother did, which I only played for Madden mostly), so I'm not a fanboy of any company, though I do like Nintendo's approach.
As for attach rate, the 360 has been out longer. Being out longer, means more time for more users to buy more games. Also, lower userbases tend to have better attach rates (like the GC had a better attach rate as far as I can remember reading). The Wii's rapidly expanding userbase makes it hard for it's attach rate to go up.
I agree with your first, third, and last point. The Wii can do better than the PS2 (since the GC could do better than the PS2), but it can't do games that the PS3/360 can. Or at least, not on the same level, graphically or in AI, etc.
But you better hope that the Wii is doing more than just bringing in grandmas, because if the Wii is not competing with 360/PS3 (and like you say, not in the same market), then the video game market is heading for a crash, because my this time last generation, there were well over 30 million PS2s out there (2 years after it's launch) along with about 10 million GC/Xbox. Even with the Wii, we're just at that 40 million in roughly the same timespan (the exact amount of times are not identical, I'm in too much of a hurry to do a full out comparison), so without it, MARKET CRASH. So yes, they are competing.