Hi, I'm Ukrainian Catholic, born and raised, but I am not familiar with the practice of other Eastern Catholic churches on the rosary; only mine.
Like all kids in my Ukrainian Catholic Eparchy in Toronto, I went through a First Holy Communion around grade 2 where each child was presented with a rosary and prayer book. We all had rosaries. I never thought of it as a latinization but rather accepted it as a part of being Catholic and a wonderful powerful prayer - the most powerful prayer we were taught against Satan according to the promises of the Rosary. In the 1980s as a student some of our priests and seminarians organized rosary crusades and had the "Fatima prayer" attached to each decade of the rosary as Fatima was pretty popular for many Ukrainians as it dealt head on with the threat of communism and the conversion of Russia. Needless to say at this time the Ukrainian Greek Catholic Church was still illegal in the Soviet Union and hence all those rosary crusades for the downfall of the Soviet Union (which I never thought would happen :)).
If one goes to Ukraine for Marian pilgrimages to such sites as Zarvanytsia, I'm positive one will be met with a plethora of rosaries.
In my parish in Canada, our parish priest holds Friday mornings in the church for recital of the rosary by parishioners.
It's seems accepted by many, no doubt, and I, personally, see only good from this.
However, one may find and will find Eastern (and Ukrainian) Catholics who would argue this is a Latinization and one should return to Eastern roots, or at least not allow the recital of the rosary to take away from the traditional order of hours, liturgies. On this subject of Latinizations, there have been many threads on this forum, as there is a long-running thread on the Ukrainian Catholic Discussion Forum in Ukraine where one sees stricter traditionalists and those who are willing to let practices stay as they are accepted. I can see the good in both but am perhaps typical in that I want our traditional practices to remain or flourish, while I want to keep what is beautiful such as the rosary.
Not sure if this helps. Just my two cents.
p.s. Yes Mary, the Mother of God is very powerful to many Ukrainians and devotion to Her and Her Protection (Pokrova). The most popular and largest image in Ukraine's first Mother Church (St. Sophia) is the painting of the Mother of God from the 11th Century (the Oranta).