I agree with you, the most important thing is that you enjoy the game, or as you said "game must feel rewarding to play". I get my enjoyment from somekind of freedom or good story, if it's even a bit non-linear or a game which gives me tactical or stratigal (sp?) challenge.
I've currently the same problem that you had. My 500MHz, 330Mb RAM computer is not able to run new games. But actually, what I'm missing? Now, I know that there is lots of good games that I would enjoy playing, but most of the new games are more for graphics, which is not even close to the top of list of what I would demand from a game. Even if my computer would be fast enough, I wouldn't buy Doom 3 because I have played it once, and IMO the only good thing in it was graphics. It's missing all the things what I want from a game. It doesn't have (even moderate) story and tunnel-running is far from freedom. As far as I know the same could be said from HL2.
Of course it's not fun if I have to spend 80-90% of the playing time for doing something totally irrelevant, but IMO player should have at least an illusion of freedom of choice. Very good example is Deus Ex, which was mostly linear, but player had freedom to choose which way to reach the goal. Player could sneak past guards and try to not kill anybody, or player could just walk in with heavy weaponry and kill everybody who is in way.