The wonderful thing about x-box live is that you can tell what folks are actually playing. It is very easy to refute the idea that the majority of games played on the 360 are shooters...
Let's take a look at the data from 360voice.com...
Top 10 games of all time (most played sessions):
Gears (shooter)
R6V (shooter)
Crackdown (Sandbox)
GHII (Casual)
COD3 (shooter)
Forza (Driving)
Oblivion (RPG)
Saints Row (Sandbox)
Uno (Casual)
Texas Holdem (Casual)
So that's 33% shooters, 33% Casual, and 33% other... hmmm
How about games with the most usage in a single day?
Crackdown (Sandbox)
Gears (Shooter)
Aegis Wing (Casual)
Worms (Retro)
TMNT Arcade game (Retro)
Bioshock (shooter)
CastleVania (Retro)
Double Dragon (Retro)
Doom (Shooter/Retro)
Boom Boom rocket (Casual)
That's 33% shooter (if ya count Doom), and 66% other.
Now this data only counts folks who have signed up on 360 voice, but still... it is clear that shooters are definitly not the ONLY game in town on the 360, looks like it's about 1/3 of them.
Here's the link: http://www.360voice.com/games.asp
Don't like that data? How about Data from "Major Nelson" on the top 20 games of 2006 on live:
http://www.majornelson.com/archive/2006/12/30/2006-top-xbox-live-games.aspx
If you review that list of 20 titles, I count 7 (including Splinter Cell). Still less than half the top played games. And back then there was much less choice.
The myth that the 360 is a shooter system comes, I believe, from the fact that the original x-box proved that shooters could work on a console (via Halo). I know Golden Eye was first, but Halo really showed how it could be done in a way that even PC gamers could see was exceptional. Of course, with the success of Halo, many other companies wanted to get in on the action, so since then many, many FPS have been cranked out for all consoles, and on the x-box platform in particular.
I think the 360 is breaking open another new genre this generation of console: The RTS. Prior to the 360 there has been no decent RTS release on a console. There have now been 2 (BFMEII and C&C3) on the 360 that have been well received and many more good looking titles coming next year in the genre.
Both the FPS and the RTS are formerly the doman of the hardcore PC gamer, which probably also lend to the idea that the 360 is for them. In reality, the 360 is the rare system that bridges gap. Now the hard core gamer can play almost everything that was offered on the PC in the comfort of his/her living room on a 50" TV.
But the casual gamer can also get in on the action via XBLA and titles like GHII, Uno, etc. The numbers I presented above prove that as well.