I think you can break the question down into two separate questions. In my experience, cat owners and people in general are far more concerned about domestic cats preying on birds than they are on mammals such as mice and voles or reptiles and insects.
If the question is how can you stop my cat hunting birds there are a couple of custom-made solutions which appear to be quite successful.
Before I mention them, the obvious all-encompassing solution is to keep your cat indoors at all times while at the same time ensuring that they do not become bored by enriching their environment and playing with them as much as possible.
Moving forward, to the solutions that I have referred to above, the first is perhaps more successful than the second which is a noticeable, brightly coloured cat collar worn by your cat which will alert birds to their presence. It is produced and marketed by Birdsbesafe®. I have a full article on it and therefore I won’t reproduce the text here. Suffice to say that 77% of owners who used this collar wished to continue using it. Also, 64% of cat owners said that the red coloured version worked the best. Click on this link to go to the page to read more about this interesting device.
Another similar device, the cat bib, is worn by your cat. It alerts birds to their presence as they stalk them. This has been around for quite a long time. The manufacturers say that it is 81% successful. I wouldn’t take that as a fact and I would prefer the lightly coloured collar over this device. I also have a full page about it. If you want to read that page please click on this link.
As mentioned both these collars frustrate a domestic cat’s hunting success because they alert prey.
If you’re concerned about other animals being attacked and eaten by your cat or brought into the house you should, as mentioned, keep your cat inside all the time. Alternatively you can keep your cat inside at night, which is a compromise. Domestic cats generally hunt at dawn and dusk because prey is more abundant then. Confining her at this time will help save the lives of prey.
Other compromises would be to take your cat outside on a leash (hard work). Or you can take your cat outside in a cat stroller. I have done this and it is just fine and indeed it can be fun too. You can build a catio for your cat(s) as a further alternative. All of them are basically about confining your cat to the indoors and then doing your best to enrich their lives in other ways by supervising their exploits outside the home.
Of course, all these efforts to curb your cat’s hunting affects your cat negatively. This is because the major emotional driver of the domestic cat is to hunt. Their whole life is built around it. They are finely tuned predators and motivated to hunt. If you remove this reason for their existence you naturally remove a large part of their life and lifestyle. This is something that must be taken into account. It is a balancing act as to whether you prevent your cat hunting completely which pleases you and wildlife against which you decide how much it affects your cat’s welfare and health.