Tangoitaliano
I think you are 100% right - this is the answer one would expect.
The Frogs are quite protective of their domestic business - they want to make it hard for pilots to operate F-reg planes on FAA licenses. The CAA, to its credit, has not done this.
However both would like to kick out N-reg planes, and flying an N-reg is the only way to do it properly
Incidentally, there is little point in doing a
piggyback FAA PPL nowadays - unless you never wish to go beyond that (i.e. do an FAA IR). This is because it is all but necessary to go to the USA for any FAA checkride, so it is highly sensible to do all required training (to reach the FAA IR test standard) out there, which means one may as well do the FAA
standalone PPL out there while at it. 2-3 years ago this wasn't the case; one could have done the FAA IR wholly in the UK and a lot of people did the FAA piggyback PPL because it was easy. Now it's pretty pointless.
It's a good idea to take a long term view on this stuff. Unless you really never ever want to do an IR, it's best to do a standalone FAA PPL, you can then fly G-reg on that worldwide VFR, and then you are all set for the FAA IR.