OP, I started a thread on a different forum with word-for-word the same title! A lot of North American people use that forum and I'd read so many stories of people wandering into shelters and choosing their cat. I assumed it would be the same here, but I was SO wrong.
We found it really difficult to adopt. Most shelters had none or very few cats available, often they didn't reply to enquiries. We had a particularly poor experience with the Cats Protection League, who seemed really incompetent and disorganised and one of their staff said they're STILL not adopting out any cats due to covid. (Sorry if anyone here works for CPL: maybe other branches are more functional.)
We weren't fussy about age/breed/colour/sex/disability. Our only criterion was that we wanted an affectionate, people-centred cat who could be a genuine companion for our (secondary-age) children.
Boy was it hard...
Eventually I applied to the RSPCA to adopt a sweet old girl. Got turned down, BUT the nice RSPCA lady said she had a litter of kittens, two of which already loved being with humans and napping in laps. We passed the home inspection thank goodness, and are due to collect the kittens in three weeks' time!
So yes, it is really hard to adopt a cat, at least around here.
Also I got flamed on the other forum where I posted about this. That was odd. I think some particularly full-on cat-lovers took exception to me saying I wanted affectionate cats. But I'm after a pet or two, not planning to set up a sanctuary for traumatised felines.