no you should not glue over cutback adhesive.

it may not hold a bond well.

plywood is a great solution.

So is surface prep to remove the cutback, but cutback can seep into the porous concrete, which can also effect the bond of the new adhesive and it have less strength, and they might not be able to get it all up wich means less bonding strength.

Plywood curcumvents the needed adhesive removal and does away with the unknowns.

The installer that saysd he did this dozens of time without issue it taking a risk with your floors because adhesives specifically state that they can not be installed over old adhesive. He is also a corner cutter and probably a lower price.

Whereas the other isntaller is giving you a realistic proven porofessional solution.

Why take the risk ?

If it fails it will cost you more.

Remember the isntaller only has to guarantee the installation for a year or two.

What happens in one year if it starts losening and popping up ?

You live with it or pay to have it fixed or replaced.

Go with the plywood.