You call it flexibility, but what remains of Christianity today, that was there when Jesus Christ PBUH was actually here? Do you think those Christians then would even recognise these people today as Christians? Women priests, homosexual priests, gay marriages etc etc. Many things that Christianity forbids and condemns are taking place freely and openly now, and just because the vast majority of Christians convince themselves it's ok, quite a fair amount know it isnt
You think those things are only present in 'modern' society?
They were evident in and around the society of the early Church too, so yes they would wholly recognise us a Christians.
Anyway those issues do not define Christianity in any way, shape or form, only the belief that Jesus died once for sin, was raised to life and that he is Lord.