Does this tub repair method sound legit?
I know nothing about this concrete thing, but I don't see how that would be an improvement over just cutting away the bulge and then dropping in the insert... and if you're concerned about what's supporting the tub, I'd suggest the right answer is to pull the whole thing, examine/reconstruct that appropriately, and put a new tub in.

STRONG recommendation to get a second/third opinion and estimate.

This is almost certainly not covered by insurance -- ordinary repair is the owner's responsibility -- unless you have a "home warranty" policy in effect from when you bought the place.