It isn't a matter of how they have been treated but how they feel about themselves.
They want to have their own Head of State.
In the US every child theoretically has the ability to grow up to be the nation's Head of State. In Jamaica they can't - and what is more that Head of State doesn't even live in Jamaica - that is the problem - having a foriegner who visits about once a decade as the Head of State.
They aren't 'part of the crown'. Jamaica is a separate realm not part of the UK at all. Since the UK moved into the EEC, as it was then - now the EU - it has had less and less relevance to the other realms - a number of which have raised the spectre of becoming republics, many of which have done so. When the Queen became Queen there were about 50 realms but now there are 16 and by the time William becomes King there will probably be single figures.
Australia will probably become a republic shortly after the Queen dies and I say that because at the moment it appears that both political parties have decided to put on hold the question of a republic for the rest of the Queen's reign.
I do think Canada may end up being the last overseas realm.