I have used veneers that I have stained... I also tried lacquers and enamel...
With veneers, you must use wood conditioners before staining. Wood varies in density. Denser sections will not absorb stains to well. This ends up with blotchy finishes. preconditioning eliminates that.
lacquers have the highest gloss. It dries rather quickly but takes forever to cure. I painted a box with lacquer and let it dry for a week. It felt perfectly dry but when I sanded it, the paint was coming off in small chunks. The underlying lacquer did not cure enough. Apparently it can take a long time to cure.
I have the best results with enamel. It dries a little slower than lacquers but cures faster. I can sand it the next day with no issues.
If you are finishing MDF and what to paint it with lacquers or enamels, you have to seal it with Zinnser BIN Shellac based primer. You apply it with a roller and it sands down real smooth. In fact, if you like a white sub, you are done. It is that smooth. If you use other enamel primers the MDF absorbs it and leaves a fuzzy finish. Even after a few coats. The Zinnser is the best.
I don't like latex. It does not finish smoothly as compared to enamels and lacquers.
If it's not a BIG screen, it's not a theater...