My stealth sat a couple years and the pain has paid the price.

It has 3 problems.
Small chips and surface scratches. I figure I'll use a hard wax or mirror glaze to soften the effects of the bigger ones.

The other two problems are contaminants stuck to the paint and dulling of the clear coat.

Nashville has some nasty air. Over time a black sooty looking coating covers anything left outside a cover. Darn near have to sand it off.

Bug and tar remover has little effect. I have been scrubbing with nylon brushes but some spots are really stubborn.

Any preferred products or techniques that don't break the bank?

I'd clay-bar the thing 1st, then compound it with 3M Finesse-It and a cutting pad at fairly low speed.
Then polish with swirl remover and a milder pad, then wax, and not a "cleaner/wax".

Or, if you're not comfortable with or practiced at the above, find a good detail shop - see how their black cars turn out - and let them deal with it.

Wax and grease remover is safe and should take off road tar and anything else on the surface.

Clay bars are also a good tool.

If you know how to use a buffer then it sounds like that's what it needs. If you don't feel up to it a detail shop will do it for a few $$$