Unfortunately, we are living in a world where we need to treat every third party application that we run (and in some cases, even the operating system itself) as a hostile attacker.

Gone are the days when you could download a software, run it, and trust that the developer isn't transmitting everything on your system back to their servers. Developers have poisoned the well and repeatedly demonstrated that they will run roughshod over the user's system, send telemetry back about everything the user is doing, enroll them in A/B experiments they didn't ask for, and take advantage of every resource, every peripheral and device, and every bit of personal information they can get their hands on.

Unless you are 100% running open source software that you have personally vetted, or that you trust that the package manager / maintainer has sufficiently vetted, you can't copy a password into the clipboard with any kind of confidence that other apps aren't sneaking a peek.

Let's assume I have control of my own computer. On this computer with Linux and X11, how do I restrict programs (such as Zoom client) from accessing the clipboard?

What are you going on about? Why shouldn't there be some temporary authorization prompt for clipboard access? You could even wire it up to control-V somehow such that the same keystroke would grant temporary access and communicate a paste to the application. Nobody is talking about making the clipboard unusable.