Keep in mind that X11 is a protocol, so the client might not
be running as your user on your local machine, it could be a dedicated machine that's only running the client.
In this case, again it's not important because in our timeline X11 is old, you might proxy the clipboard feature, with a trusted and untrusted connection, the untrusted connection needs to be careful because it's exposed to arbitrary nastiness from potentially hostile untrusted clipboard-using software - the trusted one talks to everybody else. So an example is you might decide to sanitize text, strip out invisible control characters, and exclude "rich" text formats that might conceal attacks. Or you might allow some images but only after previewing them and constraining their properties, no 18GB GIFs please, yes it's technically possible to encode a huge truecolor image as a single GIF no I don't want that in my clipboard.
Is this something we should try to implement? Probably not, but in a world where people try to kite surf across the English channel it's nowhere close to the craziest hobby.