GC's controller is great for FPS'. Take TS2, the best FPS on GC. When I used to play it I liked the C-stick for aiming while moving. If the C-stick was like the Left analog stick, I wouldn't like it at all. Back when I first got TS2, I think I didn't want to wear out my already worn out indigo controller, so I tried customizing the D-pad for move/turn. It was hard to get used to at first, but after you play a while, you like it more than you think you would.
This way, you can save the left analog for games that require it. (but I hate all games that don't give you the option of using the D-pad for character movement. hintTS3hint Damn Free Radical and EA to hell!). The left analog feels ok, but I liked N64's better (felt sturdier compared to GC's analog on a ball). The Z button isn't as bad as people make it out to be. Sure, at first it feels too hard and takes more pressure to press than the L/R buttons, but after a while it loosens up. Then there's the people that (I'm guessing here) try and press it (Z button) in the middle or left, when it works way better if you press it on the right side (where the little dot is). It isn't a shoulder button people, so stop acting like it is. You're not supposed to rest your finger on top of it, like the L/R buttons or the PSX/2 shoulder buttons.
Games I have that let you use the D-pad for character movement are: TimeSplitters 2, Beach Spikers, Viewtiful Joe 1 and 2, RE:0, RE1 (although when it came out I went through it lots of times using the the default control scheme, it rocked using analog! RE: CVX, RE4, Metal Gear Solid: Twin Snakes and NHL Hitz 2002.
The worst game that requires you to use the left analog stick is Animal Crossing. They don't even give you an option to use the D-pad. That game is a Waste of analog. Sure, I was addicted to it back when it came out and after many months of doing the same shit over and over and over, I finally traded that crap in, but got ripped since I didn't know about the good trade value at GR yet.

MGS:TS' control scheme took a while to get used to. Its control scheme is probably the most difficult to get used to on GC. So, yeah, the GC controller is the best out there.
Although I sometimes wonder if the PS2 analogs are designed so loose as to make them last longer. Now, I know that sounds dumb, but maybe they would last longer that way. I've never owned a PSX Dualshock before, since I didn't need one for PSX. I just used the D-pad for all the games I had.