Some do, some don't - and certainly they did not used to do until a few years ago - many is the high note I have hit with a full bladder! Improves the tone I can tell you!
When did you last visit a CoE church, Luckygirl? In spite of the inordinate cost of installing a loo in a Grade 1 listed 11th century building or something equally problematic, a lot of churches have done it. And perhaps the vicar wasn't so much trendy as defeated by there being no drains within half a mile of the building. Or the impossibility of digging up a 14th century floor. Or the inadvisability of disturbing the remains of the parish ancestors!
What a Cheapskate! We have one, its grand opening was only a year or so ago - but you can sit down on ours! The embarrassing part is asking for the key.
A local trendy vicar decided to install a composting toilet in a little shed behind a church - the opening is just a slit and is fine (or relatively) if you just want to pee ............ I will say no more!
Our local C of E church has one loo, so must buck the trend. It's in the dark depths beyond the altar, through changing rooms, behind screens, velvet curtains and the like. My DC always used to want me to take them after Sunday School, and I had to half close my eyes, in case I stumbled across one of the clergy or choir in a state of partial undress.