Going back to the OP…
Whether or not Eastern Catholicism is a “bridge” or not depends on many different assumptions.
First, from the perspective of the ecclesiology that was prevalent in Rome from, say, 1400-1960 or so, yes Eastern Catholic churches were to be bridges, in the sense of being means/fruits of attempts to get the entirety of local Orthodox Churches into union with Rome. This becomes more the case when you view the thought process of contemporary Latin theology/culture as mandatory and try to impose those ideas on the Eastern churches (Uniatism, according to both the definitions of Father Cyril Korolevsky and the Balamand Agreement) - where “Eastern Catholicism” is really just an externally Eastern form of Latin Catholicism.
Second, the two cases I’m most familiar with (The Union of Brest as discussed in Father Borys Gudziak’s book - which is IMO mandatory for any discussion of the event- as well as the Melkite reunion of the 1700s) - the purpose, from the Easterners coming into union, was not to serve as an ecumenical bridge, but to preserve the integrity of the “eastern” religious culture in the face of challenges from all sides (Brest) or as a show of support to a particular Patriarch and his pro-Roman stance (Melkites).
Third, yes some folks in contemporary Byzantine-rite Catholicism like to think of themselves in terms of the ecumenical separation between Rome and Orthodoxy, and hope that their existence can help serve the cause of reunion. Maybe so, maybe not. Many are pessimistic since the Orthodox aren’t amenable to our current status in relation with Rome. I personally believe the best we can do about this is to educate Latins about the Christian East, which I think we can generally do as long as we avoid directly and immediately challenging newcomers on controversial issues.
My biggest problem with the “bridge church” idea is that it defines Eastern Catholicism functionally. IMO, we don’t exist to serve a function in East-West relations, we exist as a church of the “Byzantine” tradition: the elect of Christ united into one body with the deacons, the presbyters, and under the successor of of the Apostles, the bishop. We have all chosen the “Byzantine tradition”, either as adult “converts” or as those who continue to worship in the tradition of their fathers. Our choice implies that there is a divine wealth in the “Byzantine tradition” of Christianity; our job is to find this and to live this.
Markos