So for example the US renders your UK licence valid with the paperwork you do at the FSDO before they let you hire the aircraft.
Not really correct.
The FAA does not render CAA licences valid in any ICAO sense.
The holder of an ICAO PPL is deemed to meet the aeronautical experience requirements (with certain caveats) for the issue of a self-containd FAA Pilot Certificate under Part 61.75.
There are plenty of examples of countries that will simply render an ICAO licence valid (Bermuda would be one such), but the USA is not one.
2D