As a matter of humane stray cat population control, cats are neutered and returned to their original territory. More info.

Successful results with TNR, a case study: https://www.mdpi.com/2076-2615/7/11/81/htm
Review of TNR as a means for population control: https://escholarship.org/uc/item/6z98577x
Ethics of TNR: https://www.frontiersin.org/articles/10.3389/fvets.2018.00341/full

Estimated to be more effective than neutering at controlling the population, are vasectomies and hysterectomies:
http://veterinarynews.dvm360.com/tufts-researchers-say-tnr-not-most-effective-method-control-feral-cat-population
The original paper here: https://avmajournals.avma.org/doi/abs/10.2460/javma.243.4.502

Many that oppose TNR, are wildlife conservationists and birders, who produce research articles such as this one:
https://www.mdpi.com/2076-2615/9/4/171/htm
What the anti-TNR article fails to account for, is that there aren't sufficient homes for all strays, and that some feral strays cannot be tamed. This is overlooked with the motivation of protecting wildlife. TNR is a cost effective program at reducing stray population. Given there are not enough funds to house all strays in humane environments, the “correct” solution appears to depend on which animals you choose to protect. However, this assumes stray cats live in a wildlife setting. Where do stray cats live? They live around humans, which leads the the next section of this article, the geography of cats.

Stray cats are commensal with humans, in that they thrive among human populations. Wild areas are mostly free of feral cats. Since they are not the top of the food chain, cats don't survive well without the protection that humans provide by keeping larger predators away, or by providing sources of heat during cold winters. “Since feral cats live primarily as scavengers in human settlements, the birds they kill are very likely to be the commensal species that bird conservationists hate anyway.”

In Australia, because they are killing all the native predators to protect livestock, ones that keep feral cat populations in check, like dingos, the feral cat population is thriving in the outback. “While feral cats are a big issue (for wildlife conservation in Australia), according to CNN, the government has focused heavily on them over more “politically sensitive” issues like habitat loss caused by urban expansion, logging, and mining.” 1

“In all likelihood, nuisance wildlife trappers killed three or four times as many cats as USDA Wildlife Services, perhaps half a million or more.” https://www.animals24-7.org/2014/11/14/record-low-shelter-killing-raises-both-hopes-questions

Because of nuisance control, TNR and mandatory neutering upon adoption of cats, there are some areas where stray cats are going extinct.

The unintended consequence of a lack of stray cats in certain regions, or a lack of non-neutered family pets, is that the local demand for cat adoptions is unable to be met. Local non-profit organizations are in competition to find new cats for their shelters, to meet demand, and they avoid the topic because it can reduce their funding.

Another consequence, is that the adoption requirements in those areas can be increasingly stringent, requiring references, proof of a pet allowance in the lease, and even a house visit. Adoption fees are also increasing. Kittens must be adopted in pairs, which is good practice, to avoid “single kitten syndrome”. However, that they have the ability to command such a requirement, is telling of the demand. If there were too many kittens, they would be striving to place them in any way they could.

If stray cats go extinct in certain regions, breeder cats would be the only available source. Except that stray cats are being transported from regions with higher cat populations. Even if they don't go extinct, the population in many regions have been driven down and create a situation of inbreeding. Stray cat populations depend on diversity.

Meanwhile, a large percentage of cats at shelters are “at risk”, due to behavioral or medical issues.

Petpoint Industry Data keep statistics on an aggregate of animal welfare organizations.

Many will not know the joy of having a cat as part of their family. Not only is this true for cats. The same is happening with dogs. This article describes a family's search for a farm dog: https://ordinary-times.com/2020/05/08/let-slip-the-dogs-of-quarantine

Aside from official breeders, there are persons who don't neuter their animals by adopting from others that don't neuter. These “unofficial breeders” create an underground pet market, but I don't know with what prevalence. Likely more prevalent in poor neighborhoods.

https://fixnation.org/topics/caregivertips

My cat was beaten to death by a female neighbor with bird feeding stations and a small dog. I always wondered why my cat stopped eating cat food, until later I realized, that he may have survived poisoned cat food. A year or so after his death, I was feeding a stray mom cat with a litter. I found the runt of the litter beheaded in front of my living room window.