I have brought back many paint finishes by using polishing compound (and/or rubbing compound for more difficult inperfections in the paint).
The process needs to be done by hand (no machines). You will not end up with a "perfectly flat, show car finish" but you will likely end up with restored original paint. The problem with machine refinishing is that it tends to cut the high, but because of inconsistent paint film thin areas can be cut into the primer coats. Your paint is original only once and by hand (cutting and) polishing one can control the amount of cut and not wear the paint thin.
The process I use is akin to wet sanding using heavy felt glued to equal sized pieces of truck tire(inner) tube,and then optionally a smaller square of indoor/outdoor carpet, a very flat palm of a hand and adequate amounts of (cutting or) polishing compound and water. After the (cut and) polish the area (I do about a square foot at a time) is cleaned well with clean terry cloth and finally waxed with high carnuba content wax.
Others may have similar processes, all of which take time and attention to what is being done (removing a very thin layer of paint).