peter - I agree with you re FAA accepting ppl icao training (although I think there are issues with IR training ?). The only DPE in the UK appears to disagree with you... so we are back to 'you can only work with what the other party is comfortable with' . There are a number of other "FACTS" on his website that are actually Opinions but the one that disagrees with you is as below. I think it is an over zealous interpretation of 61.41. Surely it actually means they can sign your logbook to show the training given, but that they cannot perform other duties required under CFRs ie BFRs for example. It does not say the bit in bold below...
NON-FAA Flight Instructors - The FARs do not allow for the core training to be done by a non-FAA instructor and never have. Everywhere the FARs refer to an authorized instructor it means a current and authorized FAA instructor. Although CFR 61.41 indicates that training given by a non-FAA instructor MAY be credited toward the requirements of a pilot certificate or rating, it only authorizes a non-FAA instructor to give endorsements for training given toward the proficiency level required for the flight test and NOT the minimum core hours required to be given by an FAA Instructor! Furthermore, if the non- FAA instructor cannot give the same training for the award of the equivalent non- FAA license or ratings then he is not authorized to give FAA training for FAA ratings or licenses.