This old adage still exists today - “train where you plan to fly”.
I would only suggest going down the FAA road first if you can work in the US or don’t wanna work in Europe, but jobs overseas for zero hours rated guys are slim, so its Europe like you said is where you wanna end up.
The cost saving in the US aint what it used to be and schools are as expensive over there that they ever been, plus when you come back to convert everything you have to dump a lot of what you learned over there in the theory side.
Learn in Europe.