How do I remove a kext added by 3rd party software?

Any older third-party kernel extensions that remain in /System/Library/Extensions are there to stay as this is now on Ventura's read-only protected System volume. Neither csrutil, nor root will work there.

Once you authorize an app, you authorize anything from that developer.

To see what you are going to install, or what you have installed, you can use one of the apps from here: https://mothersruin.com/

Use Suspicious Package to inspect an installer package. Use Apparency to inspect an app. The Archaeology app is a new one. I have to download that one and try it out myself.

There is another app you can use to see what software is installed and running in the background, but I wrote that one so I can't give you a link to it here in the forums.

Can you clarify that a bit? Which specific software are you asking about?

Unfortunately, this has gotten very difficult and confusing over the past couple of years. You specifically asked about a "kext". But that term usually refers to a "kernel extension". Apple confusingly calls this a "Legacy System Extension" or sometimes just a "System Extension". To uninstall a kernel extension, just drag it to the trash and restart. Easy peasy.

The problem is that Apple recently introduced a new type of "System Extension". These new extensions can only be removed by:

1) The software that installed them, or

2) By your dragging the containing application to the trash.

Otherwise, these new system extensions are baked into the operating system. If you try to remove them using any other method, not only will you fail, but you'll be unable to ever remove it again. You will have to erase the hard drive and reinstall the operating system. And you can't do a full restore from backup either. You have to make sure to avoid restoring software, system settings, or "other files". And don't you dare try an "app zapper" or "clean up" tool on one of these.

It is actually pretty easy to uninstall these by dragging the containing app to the trash. Unfortunately, said containing app is often located in some hidden directory. It is virtually never in /Applications.

Developers can also provide uninstaller to remove these extensions. But these days, they generally aren't doing that any more. Or if they do, their uninstallers simply don't work. If they are flat-out refusing to help, then that is probably the case.

There is another, super-secret method that is intended only for developers. You can find it on the internet but I can't mention it here in the forum due to security reasons.

So, no, you aren't missing anything. Always double-check when someone asks for extra permissions.