"And now that is no longer true? It's part of what has to accept when one submits to the authority of Rome, isn't it?"
The Catholic Church since Vatican II has realized that one can only be Catholic by being Byzantine, Latin, Maronite, Coptic, etc. and one cannot artificailly seperate the former from the latter. Admitting Rome's primacy and authority should not entail adopting Latin theology or praxis, although admittedly this has occurred. Some still believe Catholic first, (fill in tradition) second, but slowly we are changing this attitude.
"Doesn't Rome have to approve all those transfers from one Rite to another? From one sui Juris church to another? Seems it's a heck of a lot easier for a Byzantine Catholic to become a Latin than vice versa."
One must distinguish bewteen attending a parish of another Church and officially transferring from one Chruch to another. Any Catholic can attend a different Catholic Church's parish, even become a parishioner it does not affect their canonical status in or obligation to their particular Church. And this is what has hurt my Metropolia especially. Our people do not officially transfer from the Byzantine Church to the Latin, they simply join the nearest Latin parish. Rome is not involved, and most Latin pastors are happy to get a new parishioner and don't encourage them to travel to a nearby Byzantine parish or start a mission.
As for officially changing from Church to another. Rome is no longer involved as long as the transfer is between Churches that coexist in the same territory. So a Latin could transfer to the Byzantine, Ukrainian, Melkite, Romanian, Maronite, Syriac, Chaldean, Armenian, or Syro-Malabar Churches since they all have eparchies in the US. If one wishes to transfer to a Church that does not have jurisdiction in a territory Rome's approval is needed. Transfers from Latin to Eastern are quite easy and I have never heard of one being refused. My own took about two months. Transfers from Eastern to Latin are usually not approved.
Fr. Deacon Lance