You don't need an "FAA" instructor. The FARs accept non-US training wholly. Countless people - myself included - have done an FAA PPL, wholly in the UK, using their UK PPL training logbook entries towards any required logbook entries. And countless more people have done licenses and ratings in the USA for which previous UK and other ICAO training was accepted without a question. I have never heard of anybody whose non-US training was not accepted in the USA. It has been normal (no idea if it is now) to do an FAA PPL in the UK with just a few hours' training with an FAA CFI and the checkride with a DPE, so the 45 hours of UK training (in a G-reg C152 with a UK/JAA instructor, etc etc) was wholly accepted.
The only thing for which you need an FAA instructor is the last 3 hours (within the 60 days preceeding the checkride, etc).
So,
training towards any FAA license or a rating has never been an issue, in Europe - except for those who don't know the options
What has always been an issue is getting the
checkride, and that part of it has been good at times and bad at other times, with many people having trained up here and then found that the checkride option they had lined up has vapourised.
This is why I recommend anybody wanting FAA papers to get their ducks in a row over here (medical, exam, etc) and then
go to the USA and do it all out there in one concentrated stint. You benefit all round, from the well organised training at a proper school, from the lack of distractions, and you get cheap flying compared to here. The obvious exception is if you really want to do it in your own aircraft...
This is an area where a number of people posting on forums have received some very nasty threats so everybody who knows anything about it will always just be skirting around the issues when writing in a public forum.
Interesting point about not passing a BFR.
AFAIK an FAA CFI does not have the authority to revoke your Private Certificate (an FAA PPL) after a flight. But he doesn't have to sign you off...