I did have to ask around for a bit as I wanted to adopt indoor only cats for safety reasons (very aggressive local un neutered tom who nearly killed a couple of local cats, a car repairer very close to the back garden so things like anti-freeze all over the place, and roaming gangs of feral teens). But I found an independent but very well respected rescue who were happy to let me have the 2 who chose me, and a third a few years later as a companion as one of the first pair had to be PTS. The lovely lady who runs the rescue was particularly happy I took them as they had been there a year as nobody wanted a) a bonded pair, b) young adult or c) black cats.
I completely agree with @Happyjoe above with the reasons why many people are refused e.g. the insistence on one rather than 2, the colour thing, has to be a kitten they won't accept a young cat instead. Then you get the people who HAVE to have a kitten right now, maybe in the depths of winter which isn't kitten season so no there are likely to be very few around at the time of year from ferals or strays. People seem to forget that a cute kitten becomes a young cat in a very few months then you have another 18-20 years of an adult cat.
I can understand rescues not wanting to rehome in December either, as giving an animal as a Christmas Present is rarely a good idea especially at a time when there's all the excitement going on.
Did you know that some people won't have a black cat because they tend not to photograph well for social media? But that's OK, because come late September & especially in October some sick and twisted people get black cats & pretend to be Satanists!
I told the story above of my darling Tobias. They would have chosen him as a kitten because he's one of the prettiest cats I've ever seen - a rich brown tabby & white, almost symmetrical and amazing markings on his very cute face. But as soon as he got past the cute little kitten stage he was thrown out on the streets. That's another reason why some rescues are very wary & careful about adoption.
People have said above that they don't want a cat with all the potential traumas it may be carrying from it's past life. That's fine, there's plenty of people out there like me who have the love, time and patience to overcome their pasts. But whenever I've been there's also been plenty of friendly cats who love people e.g. had to be rehomed due to their previous keeper's ill health, death, other change of circumstances.
By the way @Millytante , try having a quick Google about whether it's better for a cat to have a litter before spaying. You'll find it's near enough 100% saying this is an old wives tale and the real situation is the opposite.