What would the recommendation be for just playing the Donkey Kong, Pac Man, Asteroids, and maybe Burger Time?
Asteroids plays with buttons only, so that one's not relevant when it comes to joysticks.
DK, Pac-Man & Burgertime are all 4-way games, and will play at their very best with a true-4-way stick. That's a stick with diamond or cloverleaf shaped restrictors. Next best would be a digitally-restricted stick like a 49-way or a U-360. Playing on an 8-way like a Competition would be your
worst option for those games. They'll still play, but you won't get as far or score as many points. As for setting the sensitivity for those games, I like it set pretty tight on 4-way games, but it's a matter of taste. It's certainly not as crucial as it is on Robotron.
What it sounds like to me, is you're searching for the magic joystick that's the "best" for
all of the old classic games. Everybody,
everybody here has been on that search before. Sadly, there just ain't no such thing. Different games will each have their own "best" stick. The stick that's best for Robotron will absolutely not be the best for Donkey Kong, it will in fact be sub-par at best on Donkey Kong. And you would really, really, really hate using a Donkey Kong stick for Robotron.
Your options:
1. Go with the stick that fits the most important of your old favorite games. Other games will suffer, but you'll get the very best performance on the games you build for. You can always use a different stick on your *next* cab.

2. Make the best compromise you can. Switchable sticks like the Mag-Stick or Prodigy are one route; 49-way or U-360 would be the other. You'll get the best all-around gameplay this way, but fewer games will have the "best" stick for them.
3. Put more than one stick on there. Adding a dedicated 4-way for the 4-way games can free you up on your 8-way choices.
4. Do swappable/modular control panels. You can swap in any stick you like if you go this route, but it'll be more work and more $$.