It is a requirement that training be given by an "Authorised Instructor" as defined in 61.1b(ii). A non FAA instructor does NOT qualify under this definition (I called the Van Nuys FSDO and got it direct).
Therefore when the individual turns up for his last 3 hours, it is touch and go whether the FAA instructor would be prepared to put his name against any prior training (as you indicate above) simply based on log book entries.
The above is thankfully incorrect. You need to ask somebody better qualified for a ruling, and such a ruling has been given plenty of times by the FAA itself.
It is absolutely routine for previous ICAO training to be accepted towards FAA licenses and ratings, in the USA. That's a fact going back years.
If it wasn't, you would have to do everything all over again. A CAA PPL holder with 1000 hours would have to do a whole new PPL in the USA, every lesson... etc
This is the way it works in most places in the world; even the ultra blatently protectionist JAA accepts FAA flight training to a degree (for the PPL they make you sit the exams; for the IR they make you fly 15hrs and do all the exams).
The trouble with aviation is that asking one interested party about it is a bit like asking a turkey whether he likes christmas (do Americans eat turkeys for xmas?).
I have spoken to many US flying schools about this, and many other people too, and the FAA itself, and universally they accept previous training. The "last 3 hours" bit I agree with you but the reason is IMHO a
practical one.
This method has been regularly used by less than honest FAA training outfits around the UK to generate extra revenue; typically they say (for the IR) that you must fly 15hrs min with them before they will send you off for the checkride. It's all b*llocks but businessmen who have money but don't have time to queue up at the US Embassy have little choice but to pay for the convenience. Personally I think they are wasting their time; they would be better off going to the USA for the whole lot, and nowadays it's very hard to get DPEs to come to Europe anyway.
The thing I would say is that some sort of authentication of logbook entries might be asked for (but never has been
AFAIK). That's why, right back to the very first instrument lesson I had in the UK, I got the instructor to sign his ID next to each lesson. This is particularly advisable if the lesson doesn't lead to anything; for example if your logbook shows instrument training but you never did complete any instrument-type Rating, then somebody could question the authenticity of the logbook. But if you did get e.g. the UK IMC Rating (and turn up in the USA with the CAA certificate) then they can hardly question that you received the training (which for the IMCR happens to be 15hrs; same as the min logbook requirement for the FAA IR). Of course the FAA IR requires a lot more training than the IMCR but we are talking about acceptable logbook entries.