What to use to encapsulate asbestos tile in basement?
If the tiles are 9" x 9" then they are probably asbestos. If they are 12" x 12" then they probably are not asbestos. However, the mastic could be asbestos for either. So, you could test, but there isn't a necessity to do so. I agree with you Inspector and leave them in place. 

There are many problems pouring a concrete floor on an existing concrete floor (moisture issues, height issues at doorways if entire floor level is not done, reducing height in room, etc.) and I don't think it's necessary. You can install a new floor covering without pouring a new floor. 

Regardless, if the tiles are secured (solid) to the floor, then you could prepare the broken/missing tile areas and install a new material over it without pouring a new floor first. If the tiles are not secure, then installing a new engineered wood floor on moisture barrier (polysheeting) on your "prepared" floor would be best, because you don't have to worry about the material "telegraphing" every defect through the new material. It's easy to install (snaps together) and you could probably do it yourself.