Probably the simplest thing then would be to alias brew install to something like spctl —master-disable; brew install $1; spctl —master-enable

Given how ubiquitous your problem is, I would be suspicious that security alerts are going off because you have a real security problem. I've seen similar problems when a piece of malware keeps trying to inject itself into various things, and Gatekeeper is catching it. The variety of places where you're getting alerts is a testament to the persistence of the malware, and not the fact that everything is actually broken.

That's OK. If I were in your boat, I'd probably be pretty frustrated.

Does the method of right-clicking on an app, then "Open", in Finder work to tell Gatekeeper to quit complaining?

Is it possible you do this on a corp machine that has Google's Santa running & it's just a language precision issue? Google Santa will definitely prompt on nearly everything & is extremely annoying for Homebrew. Google Santa != Gatekeeper though.