Fix your IR issue and N reg would probably disappear.
That is largely true, but the long history of N-reg in Europe (and almost everywhere else in the world) means there is a lot more to "fixing it" than just the Euro IR.
The FAA has a good certification regime which has led the way in equipment and even whole airframes. A lot of N-reg planes could not be placed onto the national/EASA reg because of the can of worms which would be opened in many cases. Loads of recertification paperwork; uneconomic in many cases...
Then you have the many FAA CPL/IRs flying crew-flown business jets, not just N but also all the Cayman, Bermuda, etc who are mostly flying on FAA papers as a base, who would be facing approx 10 exams just to "fix" an EASA CPL, even if they were given an EASA IR conversion for free.
It is true that if EASA literally converted an FAA PPL/IR to an EASA PPL/IR, then the low-end (piston) N-reg scene would gradually dry up but it would take some years. However I don't believe that the low-end N-reg scene is what is driving the bulk of the anti-US emotion in Europe. I think it is the business jet scene that is doing most of that, and there isn't much anybody can do about it because of the inherent mobility of most of it.
Being on the N is a hassle in some ways: the plane is owned by a trustee, the 2-yearly static check is expensive and one has to fly to an FAA RS to get it done, one needs the DfT permission to pay an instructor, and some southern European countries screw N-reg pilots for extra money or extra long PPR requirement.
However, with the garbage coming out of EASA Part M, I am happy to be on the N-reg.
I have long thought that the whole issue is only to protect the training organisations in Europe
Correct - job protection in the FTOs and job protection in the NAAs.