Its not about the project itself,take another look from a different angle.They are not under control of America as much as one might think. The whole point of GCAP to part away from US for air superiority. Which was considering offering F35 modification right to Japan (kind of similar to Israel) if they choose American company to join their 6th gen fighter development. But they haven't.
GCAP doesn't offer enough leap in capability over KAAN to justify taking on all the headache that comes with joining the program imo. (It ain't NGAD or J36) Türkiye better off developing KAAN in blocks.
Plus they have already divided the workshare equally amone three countries (33.3%) I don't see them giving up on any.
Ask the question,why would the UK not join the US project instead?