i've played games for many years on a 13" screen in the 90's, and for a while, used my 15" computer monitor to play games via a tv hookup i used to have. and usually, it was just a small part of the screen, like 300X400 pixels. now on a 19" tv. also played a few rounds (not much, like occasionally), on a 27", but i'm not really into playing on a big screen all the time. all were great to play games on, only had slight trouble with the computer one, since that one was a bit small. so you don't have to have a big fancy tv screen to play games.
with that in mind, try the wii for a week on the 13" tv, and if you don't like it, then get the new tv. play with the size you feel good at, and don't splurge just because others have that game on the big screen. but i would say 13" would be the smallest i would go for a game.
btw, don't go too large, something about it not being good for games...