Generally, I like the idea with the user agents filtering and “block everything” rule. No need for geoblocking. Insert a comment about why this is happening and ask for it to be changed.
However, as we’re living in the real world and the authors of the respective browsers strike me as lazy or uninterested, I also bet all that would change is the user agent.
"User agent" is a synonym for "browser". When you say "user agent" here, what you really mean is the contents of the header that identifies the user agent, i.e., browser. Calling it that is a little bit like referring to Chrome's developer tools as "Inspect Element" (based on the mistake that that's supposed to be its name, rather than recognizing that the label is just a simple, descriptive verb/action).
However, as we’re living in the real world and the authors of the respective browsers strike me as lazy or uninterested, I also bet all that would change is the user agent.