In my granted limited experience with floor tile, if water gets under them the mastic/mortar lets go and the tile feels loose. If they aren’t loose I’d think your underlayment is okay. The white removable “stain” may just be some other surface spill deposit.
Since you have reattachment experience, and want to regrout anyway, you could try pulling one of the toilet tiles to check underneath. If it doesn’t want to come up easily, I reckon I wouldn’t risk cracking it, and just regrout.
Joel