Many schools will have their favourite FIC provider and on that basis schools who do FIC will favour their own product. Work for a new FI is difficult to find so a little pre-course investigation is worthwhile.
The problem with getting the Restriction removed is down to the number of solo flights available. Schools who do lots of trial lessons may not have many solo flights, so it is all dependent upon the number of PPL students who are at the solo stage and the number of Restricted FIs who need a tick in the box.
An EASA FI qualification is good throughout Europe but of little use outside Europe except at EASA approved schools where you may need dual qualifications.