Can I Polish Brushed Trim Rings?

Yes you can, but it's a fair amount of work, with stainless being more work than aluminum, but neither are easy. The first thing to do is get them extremely clean. I use 4 ought steel wool with car wash soap and distilled water. Then you have to repair any visible small dents or dings with a body hammer and a small dolly. The shinier you want the part, the more perfectly straight it has to be. This by itself takes some experience and patience. There are dozens of videos on this type of work.

When they are fully repaired, I like to start by sanding them with progressively finer grit papers. I'll start with 400, then 600, then 1000, then 1500. Sometimes I'll go as far as 2000 grit. When the sanding is done, I use an 8" buffing wheel and the correct compound, and gradually buff them to a mirror like finish. Occasionally I'll use a softer wheel for the final buffing, and a finer compound like jewelers rouge if it's something that's going to a show.
Make no mistake, this is tedious, dirty, time consuming work, and most people are happy to pay a polishing shop to do it for them in less than half the time.

As an example, about 10 years ago, I restored the trim rings and center caps for the wheels of a 67 Corvette. The parts were in decent shape, but every one needed at least minor repairs. The four rings and center caps took me a total of almost 40 hours of work to get "not quite" perfect.