Solid Hardwood, Engineered and Laminate Flooring - Need clarification on hardwood installation

I am refinishing existing red oak flooring, as well as installing new red oak throughout the rest of the living areas in my home. Most of the house has wood subfloor, however, the living room is different. I pulled up all the old carpet and discovered concrete slab with existing VCT (or possibly VAT) tile. The tile is perfectly flat and level, is glued to the cement, and has been in the house for 40 years. It is a bit crumbly around the edges of the room, where the carpet tack strips were.

My question: what can I do in the living room? Do I have to pull up the VCT/VAT (I know about asbestos, let's leave that issue aside)? Is there some sort of mastic I can apply over the VCT, then lay plywood subfloor?

I am open to gluing, floating, whatever. I just want the wood in the living room to look the same as the rest of the house, so I suppose I need an unfinished engineered product to float?

Help!

Yes you need a floating floor, but I do not know of an unfinished engineered. Maybe sand the engineered down after you buy it?