I should have explained. The floor is typically more damp and cool than the surrounding air. Carpet prevents air circulation next to the cool damp surface and the carpet will also be cooler and damper than the surrounding air. Organic stuff collects in the carpet. Mold happens. Have you considered removing the vinyl and staining the concrete? You can make it LOOK warmer.

I found some for 99 cents a tile (20 inch tiles) at a local flooring warehouse store. Actually got a little bit more of a price break because I bought the whole pallet from them! I am going to use double stick carpet tape (from Lowe's) to adhere them to an exposed concrete slab. All is hypothetical now but I hope it works! There wasn't much color selection at the store, but I figured this was a less expensive way to test the product. And then I could always go with FLOR tiles to add some diversity if the concept works well.

Seams in VCT or VAT can be almost invisible. What size are the tiles? Terrazzo is marble chips in a base of epoxy or mortar. That doesn't look like terrazzo to me and in Florida I've seen a lot of it. The colored chips do not look like marble chips. Whatever it is I'd have someone dig at it a bit before planning to put anything over it. What do you see where it meets another floor at a doorway or at the perimeter of the room?

One of my past homes had the black and green marble, hated it. I popped it all off and while waiting to decide what to put up instead I used spackle to fix where the tile pulled the drywall and then painted it. I later had travertine put in. I did a quick search and see someone painted their marble with chalk paint. I’d worry some about peel and stick for the area getting the heat, you could always buy an inexpensive one and try it right where the heat is the most significant and run your FP to see how it performs before doing the whole thing.