Gondor and Arnor a handfull

[Kuruharan]

Were there actually any High Kings between Isildur and Aragorn?

I think that the answer to this pretty much sums up everything that could be said about this. The answer is a simple no.

"So passed the first victim of the malice of the masterless Ring: Isildur, second King of all the Dunedain, lord of Arnor and Gondor, and in that age of the World the last.

That from the Unfinished Tales: Disaster of the Gladden Fields.

It worked apparently for several hundred years

It worked from SA 3320 to TA 2. As stated by Tolkien himself, the system fell apart on the death of Isildur. That is a total of 123 years, the remaining lifetimes of the founders of the Realms in Exile. The system did not survive their passing. After the death of Isildur, Arnor and Gondor only had a rather nebulous alliance. At least until Arnor fell apart and then things became more nebulous still. However, the Gondorians probably paid little attention to doings up North by that point.

also, why would Gondor wish to declare independence?

Ask King Meneldil. However, I suspect that he, like most people, disliked having anyone in charge of him. Such things are still harder to bear if you are the ruler of a powerful land.