In fact 50 years ago when all those Latinizations were common, how many Eastern Rite Churches were getting ready to close there doors permanently ? Not too many I’d wager, though that’s sadly not the case anymore.
Actually, this is a huge problem. When they were being fully eastern, few were closing their doors. When they started being western, they started losing parishioners. Why would people drive all the way across town for the rosary or stations at the Eastern Catholic church when they could walk to the Roman Catholic church down the street for it?
If you really feel that saying the Rosary in an Eastern Church somehow does harm to your Eastern spiritually, then all I can say is that it would probably kill you to endure what Traditional RC’s have gone through since the 1960’s.

What does one have to do with the other? It does not matter if others have had worse and it does not matter if things could get worse. Bad is bad and we should work to make it good.
Finally, hypothetically speaking, if I was an EC priest, and some people suggested to me that we should start Rosary devotions, while another group wanted a particular Eastern devotion, what would my response be ? I’d seriously endevour to accommodate both groups. Our Church is big enough to include quite a number of different devotions and that certainly includes Eastern ones that haven’t been in use in a while.
If someone approached my priest asking for a public recitation of the rosary, I would expect him to do the following:
-Lovingly explain that the rosary is a wonderful devotion which many people find to be spiritually beneficial.
-Lovingly explain that it isn’t in our Eastern Catholic heritage and therefore not appropriate for public prayer.
-Lovingly explain some of the alternatives available, such as the Hours, which could be incorporated if not already present.
-Genuinely thank the person for seeing a need for corporate prayer and following through with leadership and action.
-Encourage the person to take a lead in getting Matins, a weeknight moleben, or other prayer service off the ground.
-Give the person sufficient information (in word and/or book) to understand and lead the effort.
-Give the person sufficient support to lead the effort. Teach about the particular prayer during the homily, put it in the bulletin, talk about it in Sunday School, and mention it one-on-one with parishioners.
-Praise the efforts of the person, encouraging him and others to see a need and take responsibility for meeting it.
-Be punctual, present, supportive, and enthusiastic in participating in the prayer service.