Clients are not proxies. The fundamental difference is who accesses the target server. If you’re using a client (in the sense of a local program that renders your request, like a browser or a browser with extensions) then you are the one accessing the information, and once you have that information it’s hard to make a legal argument against transforming it however you want for your own consumption.

But a proxy means you request information from someone else, not from the target server, and the proxy requests from the target server. This is also fine! Except: a) the target server can absolutely say “Proxy A is not allowed to access me.” [1] and b) if the target server has copyright on the material being requested, then the proxy may be violating that copyright by making the data available for your request.

[1] there’s a weird middle ground here where if a server can tell you’re using a client it doesn’t like, then technically it could use its arbitrary denial ability to deny access on that basis, but that doesn’t seem to be legally accepted? IANAL. Or maybe that just doesn’t happen because there’s no way for it to tell what client you’re using. They certainly do try to deny service when they can figure out you’re using an ad-blocker.