For modern, see either the Catholic Cathedral in Liverpool, or Coventry. Both are wonderful expressions of faith in modern architecture.
Can I also suggest a guerilla (as opposed to gorilla) approach? Wind up the ISO, and hand-hold a few shots. This almost always fits in with the needs of other visitors (and don't forget, there are a few who go there to pray and worship: other activities should fit round these. No prayers, no need for cathedrals...)
1/60 @ f/4 and 1600 will deal with an awful lot of buildings.
Others have said a lot about processing, and I think you're right about the composition being sub-optimal. Cathedrals are one place where central composition can look really good and appropriate, conveying stillness and calm. Even if finding a central point in older ones can be a touch tricky!
So shooting straight at the altar, with the stained glass perfectly aligned with the cross could be a good way to try. It's possibly a shot to try out every time you can.
But the big problems here are the inscribed slab in the floor, partly visible; the dedication on the pew (partly visible - and there's a school of thought that individual names on such things aren't a great idea: how would you feel about your great-uncle or your grandad's name being on a website or in a magazine? I'd be OK, myself, but not everyone shares the view); and the big blank expanse of floor.
So, making the most of what's in the frame, and with crude colour and Levels adjustments, I offer a square crop. What do you reckon?