'Fabric of space' is used a lot, and seems a pretty good way to think about it. Space (spacetime) has many properties, some better defined than others.
I think for now, the preponderance of evidence for expansion, means we have to run with it.
I'd dis-count arguments against expansion based around red-shift. Many I have seen mix up doppler shift due to relativity and redshift due to expansion. I don't pretend to understand (yet!
) the detail but apparently the different kinds of redshift have unique signatures and I have no reason to doubt the experts who
do understand how it all works.
The other discussion you'll probably see around here will be about how co-ordinates are chosen in relativity - that sort of thing, and as I understand it are mainly about how we build our models of reality. I doubt that most of the people in those debates would want you to abandon the Big Bang yet.