The "old adage" as quoted by the last poster, may still exist, but is it a good one?
Why do CAE (Oxford), L3 ( CTC), Lufthansa, to name just 3, all train pilots outside of Europe?
As for "convert everything" when you come back, it should be pointed out that skill tests for EASA PPL, MEP, CPL, & FI are carried out in the USA and elsewhere in the World after completion of the approved course, so nothing to convert, just straight to licence application with the Authority. Granted the EASA IR (initial) flight test (not the training) has to be done in European Airspace, but with the new CBIR options for reduced training, the process is proving simple and cost effective for many returning home after training abroad.
I actually know someone who spent just short of $45,000 in the USA achieving their EASA CPL/ME & FAA CPL/IR, spent around $4,500 at a well known school in Exeter, UK on the CBIR portion, completed the MCC/JOC (not sure what it cost them) and managed to get an FO position with a well known budget airline flying the 737-800 with Type Rating paid by them (which is paid back out of salary as I understand it) and has a total time flying in Europe of around 5 hours. Certainly an outstanding individual who achieved this, but I hope it makes the point, it doesn't matter where you learn to fly, what matters is the quality of training. Please also remember something else - do your own research and plenty of it before deciding - said many times before by many on this forum who genuinely try to help the newbees!