Old ceiling is uneven cause it was probably installed with lathe, plaster and nails. Screwing drywall into the joists will securing it all and be a lot more flat plus no demo. 1/4 inch will do.

They haven’t a clue what they are doing. FIRE THEM NOW OP!! Any professional would have put new drywall over the old textured surface. They probably didn’t even have a stud finder. Absolutely nothing but wasted money is what you’ll have with these schmucks.