Why? It sounds gross. Simple answer, won't work without extensive floor preparation.

Won't stick for long (think days). Needs a perfectly clean flat surface to adhere to and that tile if not it. The surface will not allow the peel & stick to sit flush. Plus with all the finish on tiles will not stick either

No - not okay.

Perhaps have the floor regrouted if you can not replace the slate.

I would not do this. it will not look good.

A better investment might be a professional deep clean and see how that brings the surface up.

I bought a home with a slate entry. I cleaned it (hands and knees, scrubbed to get the dirt that accumulated in the crevices clean, then added grout where some was missing and recolored the rest of the grout. Came out beautiful.

Before and after

It doesn't appear to be just dirty, rather that somebody applied some kind of brown coating to it which has been partially worn away. As suggested above, thoroughly clean it and repair the grout if necessary. I have some slate that I don't like but can't remove. It looks a lot better when it's clean.

Many homeowners applied high gloss finishes over slate. Thankfully my slate was just dirty. Here is a link to someone who stripped the finish

https://www.effierow.com/creating/blue-slate-floors-stripping-refinishing/#:~:text=The%20first%20step%20in%20refinishing,but%20carefully%20applied%20the%20solution.