Yes. And so do the faithful in Eastern Churches. Only the Latin Church offers its faithful the host but no chalice. Don't ask me why. As a Catholic.
Would it proper for the Priest to drink the wine if the "Church" believed Jesus did not?
Yes. The chalice was offered to the followers of Christ. The Church understand it to be a sacrament. As far as the Church is concerned, Christ is God. He doesn't need a sacrament. God lacks nothing and needs nothing. Humans do.
Yes. And so do the faithful in Eastern Churches. Only the Latin Church offers its faithful the host but no chalice. Don't ask me why. As a Catholic.
That practice, as far as I know, only became popular in certain American dioceses that also, completely coincidentally of course, engaged in post Vatican II nonsense including guitar Masses, pro abortion nuns, liberation theology preachers. It is either entirely corrected or nearly corrected as BXVI imposes orthodoxy upon the Augean stables that house the USCCB.
This is rare and not the norm from my experience,dear kosta. I travel quite a bit and have gone to many,many different Latin Churches and I can say that over 90 percent or more offer the Chalice during daily Mass and I have never once not seen the Chalice offered at Sunday Mass