There should be no excuse for lippage in this day and age with the use of levelling clips it should be avoided. However if the tiles vary in width by 2mm then they could also be bowed.
The point the OP was making is he asked if a levelling system was being used, was it ok to tile over the cracks, and were the tiles ok to use. All assured replies from his tiler of choice. In the end he chose the ‘tiler’ within his budget but unfortunately it looks as though corners are cut to turn out a below standard job. When getting estimates it’s only fair to try and get like for like and in this case it was outside of budget, so the ‘I’ve done it this way for years’ won the job.
We see it a lot on here and really sometimes you have to ask, why do people pay for this standard of work.
I get a deposit, but balance on completion. The customer only pays when they are happy. Why pay if you don’t like what’s been done!!