Physical feedback is an important part of the user experience. There's no hard requirement, it just makes the device feel better to use

Yeah you are right. There are mice that don't "click". But they feel aweful.

I don't think there is something you could just easily swap, but I'm not sure.

Depending on the mouse you can replace the microswitches for clicking with silent switches

I try to click on something and it doesn't react.

Is this because I failed to execute the click (has happened, especially on laptop touchpads), or because the site is unresponsive/buggy?

As a person not too familiar with hardware or what goes into mouse creation, I have always taken the "click" sound as something that can only happen when I actually successfully execute the click. That I made those two connectors touch. (Of course, the OS might drop it and the software part does not work, but I do take it as I made all the physical things that need to happen to make a click happen.) So if I get to hear the "click" on a mouse I know for sure if I clicked or not, if it's human error or not in the problem I posed. If I don't I'm going to be clicking a lot more times to ensure I got a real click through, because I am not sure if I actually managed to click.

I feel like I have used a couple mouses in my life that don't or barely make a audible click. The important part of the click feedback to my is their being a little resistance and then it fully deforms after getting past a threshold. The click does not need to be audible.

Button or switches need to be open or closed. A spring strong enough to force the button to switch when pushed is used because if you were to push it down and let the contacts touch slowly, the contacts would bounce and you'd get multiple clicks with a single press (undesirable). Mice tend to have very little button travel which doesn't lend itself well to using other kinds of switches which allow smoother / quieter switches.

Some quieter or less clicky switches probably do exist but its at the tradeoff of greater travel distance, mushy feeling or double clicking. I recall a mouse with rubber dome buttons that was discontinued because it feels like crap ans no one bought it

Specifically for the extra buttons on the sides, I would prefer having gamepad-like switches, since the normal clicky ones don't seem to do well over there.

While an audible click is not really necessary for switches, I consider it important to at least be able to feel it on my finger. Hence the word "tactile".