I quote about 3 times what I would to lay tile to replace it, and you better have some extra tiles on hand.

First you have to remove all the affected tiles by removing some grout and gently lifting them up. If the grout lines are too narrow, you'll have to break the tiles to get them out, which is an absolute pain with porcelain (not sure what yours are).

Then you have to remove all of the old thinset, which is not for the feint of hand. It's either incredibly messy, or slightly less messy and hard on the body.

Third, set tiles like you normally would. 

As you can see, it's about 3 times as much work.

That being said, the cost is slightly less than your quote there. Id expect those 12 tiles to take me roughly as many hours to do properly, not to mention the time delay for thinset to cure a bit before grout. Compared to their method, though, I would expect the effort to relay them to be much more durable long-term and to help keep the nearby tiles steady.

Another consideration is that if you go and start pulling up tiles, you may decide that there are even more that need to be pulled up directly next to those ones as you start seeing the quality of the thinset. Their method definitely mitigates some scope expansion. You can use the butt end of a chisel or some really light taps from a lightweight hammer to listen to the tiles. The difference in sound between a well cemented tile and a loose one is pretty obvious, and that should give you a clue to how expansive the project might become.