Many sites don't contains the full content even if you do that. I'm not sure of how it works (does it subscribed to them all?) but https://archive.ph/ is a good way to see the content in those cases. But really, if you are regularly reading content on a site you should subscribe to support the journalists employed there.
Is this even legal? How can sites host other ones' content, and that's ok? - Is it fair use because it's "archiving" the web? - Is it because it's on the open web and it's public domain? - Or is it illegal, and people do it because they can ¯\_(ツ)_/¯
Is Google cache legal?
What about Internet Archive wayback machine archive?
Is deleting your cookies legal? How about spoofing your user-agent?
How about a browser plugin that automates what OP describes?
In the case of caches like Google, Internet Archive, or `archive.today` (same thing as `archive.ph`)... probably, in the USA? If it winds up in court, we will find out, eventually.
Simply reading anything on the web technically involves "making a copy" already, which is one reason it gets and has remained somewhat confusing and complex to determine what is or is not legal with regard to copying web content. You can't simply say "making a copy is not allowed".