I used to think this way, and it is a noble ideal. But in reality, it goes against everything that both Churches teach and ends up watering down both Churches.
The Catholic Church falls under the condmenation of the Council of St Sophia (879) which was co-signed by Pope John VIII, for teaching the filioque, which was also condemned at the COuncil of Blachernae in 1285 (cf. Crisis in Byzantium by Aristeides Papadakis.
The Council of St. Sophia may have condemned the filioque, but if Pope John VIII co-signed it I can't imagine it condemned the Catholic Church. Of course, no ecumenical council has ever condemned one side or the other because, by definition, an ecumenical council would require participation by all the bishops.
Now the filioque controversy is a good example of how misunderstandings arise where there are language and cultural barriers (and the irony is that we are discussing in English a misunderstanding that arose between those who spoke Greek and Latin).
As those who are much more knowledgeable than I am about history have been saying on this string, the western churches underwent a sort of filioque creep in their recitation of the creed. For awhile, the popes resisted it.
Now someone who would argue that the Holy Spirit also proceeds from the Son (as I would) might point to John 20:22-23, where, after his resurrection, Jesus breathed on the apostles and said to them, "Receive ye the Holy Ghost: Whose soever sins ye remit, they are remitted unto them, and whose soever sins ye retain, they are retained." Now that looks like the Holy Spirit proceeding from the Son to me.
On the other hand, one who would argue the opposite position would point to Jesus' words at the Last Supper, where he says, "But when the Comforter is come, whom I will send unto you from the Father, even the Spirit of truth, which proceedeth from the Father, he shall testify of me...." (John 15:26) Here it looks like the Holy Spirit proceeds from the Father only, and the Son only sends him.
Both scriptures are there, and it looks like there's proceeding and then there's proceeding. Technically, proceeding qua proceeding seems to happen only from the Father. But the Spirit comes from the Son too. What might be the genesis of the confusion is the fact that the western churches developed for awhile without much knowledge of Greek. I think the same cause is responsible for the confusion that seems to have existed, for a time, between Gehenna and Hades. In English, I would say that the Spirit proceeds from the Son, but this question has to be addressed in Greek where the rubber hits the road so to speak.
The place where it should be addressed is in an ecumenical council, at least insofar as recitation of the creed during the liturgy is concerned. This will require all the bishops from both sides, and will further require unanimous agreement among them. But nobody is authorized to simply say, "Unless you agree with me, you're not in the Church." Christ didn't set it up that way.
In order for the eastern and western churches to comprise two separate Churches, there would have to be an ecumenical council authorizing the split. That has never happened.