Lobrochta, my flooring store required the subfloor to be completely even otherwise they would have had to use transition moldings where the floor was uneven. I wanted a continuous floor (no moldings) so I removed the old flooring and made sure the sub floor was in good shape. That said, I had one area where the sub floor was uneven (one piece of plywood subfloor was 1/4 inch higher than the piece next to it) and the installer laid right over it. It created a horrible squeak and flexed when I walked on it. The installer had to remove planks, plane the subfloor, and reinstall and glue down those planks. Having gone through that, I’d be very nervous to install without removing your old tile. I’d want to know more how they do it (perhaps they build it up with mortar??) and what they will do if you have an issue. Personally, I think the right way to install is to remove the old tile and after my experience, I would push for that or find a new installer.
Q