The FAA IR will always be completely safe. The only body which can mess with it is the USA and they won't do that since most of the world flies under FAA licenses/ratings. Every U.S. airliner has an FAA ATPL in the LHS and either an ATPL or an FAA CPL/IR (like I have) in the RHS.
A particular country has some control over who flies through its airspace, but this isn't politically possible to mess with. Angola can do that for example (and shoot down anybody flying without permission) but one can't do that sort of thing in Europe.
What this comes down to in the end is control over long term parking of aircraft, according to their registration. Trying to work out how this control could be implemented and enforced is a mind bending exercise, particularly given the very low numbers involved (very low 4 digits for all of Europe). The main population (foreign reg jets etc) will escape simply through mobility - no parking control is possible on them.
There is a side issue involving the use of foreign (FAA) licenses in domestic (G-reg) aircraft (and this could be stopped easily through a change to the ANO) but this is largely moot anyway since very few pilots fly a G-reg on an FAA PPL, and anyway the UK is the only country in Europe that automatically validates ICAO PPLs for use in a G-reg.... The N-reg issue is mostly over the FAA IR and then you are flying an N-reg anyway.