Asbestos "bleeding" through the wood doesn't seem to make sense. I'd think this might be a good application for a floating floor. Perhaps some kind of laminate?

The asbestos is not going anywhere.

The tar may well start to move.

A layer of wood to anchor the new floor onto is probably going to be needed.

My home was built in the late 50s . Took up the asbestos tiles and soaked the cutback adhesive with Lestoil straight from the bottle, then used a stiff bristle brush. Yes , it was a bit costly and time consuming. But oil cuts oil- this leaves no residue because of the detergents in it.
I think the name of the adhesive was Bostick. Used that when laying engineered down. No cutback leaking thru- have a light maple shade.

I agree with mike_kaiser, floating floor (pergo type). Level last ,underlayment and a quality lament. Check out Lumber Liquidators.

There is a simple test that will determine if in fact you are facing the potential removal of "cutback" adhesive.
Apply boiling water to a small area of the area in question.
If the adhesive blisters and becomes liquified, IT IS NOT CUTBACK ADHESIVE, and can be conventionally removed.
If it is not affected, IT IS CUTBACK ADHESIVE, and you'll need some special softeners and cautions to remove.
Google cutback adhesive removal.
The product I've used was an orange oil.