At least some of these churches owe their beginnings to the political events in the Middle East of the Crusades or to the influences of various degrees of Western colonialism, so there are more parallels to Uniatism than perhaps meets the eye, even if the parallels may be more distant in time. Also, some of these Eastern Rites are so tiny that they are absolutely dwarved by the church they are supposedly competing with, and that makes the situation less contentious.
And in each and every instance, the Roman church convinced some people and or clergy to leave the jurisdiction of their church and come under the jurisdiction of Rome. Parallel church structures were set up in direct competition with the "original." Ask the Armenians what they think about the Armenians who joined up with Rome...
The only situations in which nearly entire dioceses went over to Rome was when the power of the state was there to enforce it, as in some corners of the Austro-Hungarian Empire.
It is good that the Melkites and the Antiochian Patriarchate seem to get along, but this is an anomalous situation that is unlikely to be duplicated elsewhere. Rome cannot have it both ways: they cannot both have parallel churches that compete with the Orthodox *and* expect that the Orthodox churches being "paralleled" are going to be excited about having warm relationships.
Well, the Maronites were never out of communion with Rome and the Melkites, they do owe their existance to politics of a sort. When the Saracens attacked, the Byzantines stayed home knitting and the Crusaders helped without any pre conditions so they joined up with Rome. The situation with the various monophysite "reunions" with Rome are of course another matter entirely and relatively recent as these things go. I sincerely doubt they have ever been envisioned as a wedge against Orthodoxy. Western colonialism had a terrible effect especially on the Maronites and to a lesser extent on the Melkites, with the connivance of the Vatican, but the Melkites have pretty much shed that and the Maronites are working on it.
"Rome cannot have it both ways: they cannot both have parallel churches that compete with the Orthodox *and* expect that the Orthodox churches being "paralleled" are going to be excited about having warm relationships."
What's that smell? :)