I've been a member of several different Ys. Never have I heard a shred about Jesus or God in the building. I think the Christian bit is historical. Jews in NY have the YWHA, primarily the 92nd St. Y. I have attended different fascinating lectures at the 92nd Y. It is a very upscale neighborhood. Judaism was not mentioned. My hunch is that Christian was a code word for no Jews and Hebrew was a code word for no Christians. The reality is completely at odds with their account.
I did join a Y when a teenager b/c of this issue. My assumption is that the money is used to maintain the facilities. They charge a reasonable amount for the facilities. Not cheap by any means. There are no Bibles or religious tracts aside.
Are we supposed to refuse Red Cross services in a disaster b/c once upon a time in Europe a Christian thought it was good to show a standard cross so each side would know its humanitarian principle. The Red Cross would have no role to play in war time if it were aligned with other side and delivered munitions. The Middle East has the red crescent.
Some pagan things are acceptable and other pagan things will get you disfellowshipped. We are descendants of pagans. It is our heritage.