As Patriarch Alexey recently stated publically, the practical purpose of the Balamand agreement was that it was the result of a working commission to try to address the Uniate issue. It has no dogmatic significance or authority. As he pointed out, it didn't even successfully do that, as far as real results are concerned, although the official text has Catholicism agreeing that it should be an approach left in the past.
JK is right that Vat II and subsequent encyclicals are very supportive of the Eastern Catholics, but this is primarily in the sense of confirming their equal right to existence and independence within Catholicism. The term Uniate, including its negative connotation, was coined not by Orthodox, but by Roman rite Catholics. Uniates were accorded second class citizen status for a long time. This has changed in the decades since Vat II.
What has been abandoned, at least officially, by Catholicism is the idea of using Uniatism as a bridge between Catholicism and the Orthodox Church. There is a practical recognition, I would suppose, that it has quite the opposite effect of what is desired.
Can you refer me to where Catholics are no longer using "uniatism" as a bridge? Perhaps we just don't use that term? I am not aware of this, but I am not sure.
Regards