I can pretty well guarantee you that playing thru the tv will not be a
better gaming experience. The TV's resolution just doesn't compete with your
pc's monitor, e.g. it'll look at lot "blockier" on the tv.
Also a video capture card is for bringing video INTO your pc (e.g. record
old vhs tapes etc.). You need TV-OUT to get your pc's screen to the tv. Also
note most TV-OUT implementations have some limit on the maximum resolution
the pc screen can be in order to work, usually it's 800x600 or 1024x768.
It's a hardware limitation as the tv doesn't get much past 500x400 so the
picture has to be down sampled for the tv. Also note many implementations
also set your pc's refresh rate to 60Hz (NSTC - North America anyways) to
match the tv's refresh rate.