I used to work at an open intake shelter. The answer is that we didnt necessarily euthanize to "make space" because of how the shelter worked.
The shelter had two main sections, the back, where the animals were either on stray hold or quarantine, and the front, where the adaptable animals were. Once an animal "made it" to the front they were very rarely euthanized except in extreme cases (i.e Parvo outbreak :c ) if the front got too full, we would call "no kill" shelters and fosters in the area to come pick up animals. Caveat here that we have a lot of other shelters in the area, and it may be different for places that only have 1 shelter.
Meanwhile in the back, we had to legally put animals that had been found or abandoned on "stray hold" for two weeks. After that we could choose to euthanize or move the animal to the front. During those two weeks, it usually became apparent what would happen to the animal. Some simply would not be adoptable (constantly shaking in fear, could not eat, could not use the bathroom, bite risk, etc) and it was considered more humane to euthanize. If the back got too full, well, we had to find creative solutions becuase we legally could not move animals until the 2 weeks were up. (BTW Quarantine time can vary greatly from "this cat needs to recover from its spay" to "this dog may have been exposed to rabies and needs to be under observation for months). So to answer the question: it happened less to make room, and more often because it was time to euthanize an animal we knew we were going to anyway.
The other scenario I want to mention is that many people surrendered their old and sick pets to the shelter specifically to be euthanized because they could not afford to do so at the vet. We also had to perform court ordered euthanasias and rabies tests (which requires the animal to be killed, unfortunately). Please keep in mind that when you see high euthanasia stats at open intake shelters, it is including these cases.
Side note: if someone surrendered an animal for euthanasia and we did not believe it should be euthanized, we had the right to refuse to do so.
This question seems like it was asked it good faith so I will give it an honest answer. I will admit that there were times when I wished I had more time to work on behavioral issues with an animal, or that we had the resources to treat an injury or illness. The busier the shelter was, the less time and resources we had to work on "special cases." I guess you could see that as "making room" but it was more like weighing if our limited time and resources were better spent on 1 bite risk dog who could potentially be rehabilitated with lots of hard work or a litter of 8 puppies. We often chose the puppies. I wish we didnt have to make these hard choices, but somebody had to. Working in the shelter often felt like being the sin-eaters of the animal world. Some of those no kill shelters would come gladly take our animals then turn around and call us all murders. Meanwhile our staff would give up our breaks to try and rehabilitate as many animals as we could. Shelters weren't getting any bigger, we weren't getting any more staff, and it is inhumane to let an animal starve to death on the street. I will always believe that there are many things that can happen to an animal that are worse than a dignified death. I've seen them.
Anyway, at the end of the day I wrote all this becuase I genuinely liked working there despite it all. I moved on because better opportunities for me came up, but I genuinely like talking about this stuff because I care about it a lot. I didnt work there for very long but I can try to answer any questions people may have. Disclaimer of course that this all happened at 1 specific county shelter in the US and different places will probably be different.
