" personally feel it's very unfair to say 'baby is welcome but not if they're crying/fussing'. It's a baby. It cries, not just in a worse case scenario but regularly!"
Yes, this.
One of the things that I don't really like at church (although it is meant well) is when people congratulate me at the end of a service on the behaviour of my children. It is a C of E church which has a Sunday school (which dd1 doesn't want to go to right now and I don't want to force her - dd2 is too little) and a creche (which is, erm, a bit dodgy, in the nicest possible way). So I keep my children with me and they are, so far, good. I was brought up RC where all the children, however fiendishly naughty, all go to Mass and no one made any comment on it. I don't like hearing that my children have been "good" in church because I hear, perhaps paranoiacally, that they are not welcome unless they are quiet.
Going to university, or work, with a baby that I would rush out of the room with if it fusses, would feel like that (except justified - I don't think the people in church do have a right to say children can't be there unless they are "good")
So with church I am on the opposite side of the fence - of course they belong there, they are part of our community. but I don't like having their quietness drawn attention to because it seems (if you are paranoid) to cast doubt on that.
If you think I am going to rush out with my baby when it squeaks, then you think I only conditionally have the right to be there.