I see no difference between visiting a webpage with a browser or through a program which modifies the content prior to delivery. That's exactly what any a browser plugin does.
The big difference is that if the program is run on the server of somebody else, it needs rights for redistribution (copyright) for the redistributor server, which is a different entity than you (who is running the local program).
Exactly, people want to make this a simple case (outline is really just a browser by a different name) but copyright isn't a bright line domain: intent matters and outline is just re-hosting other peoples content for broad consumption.
Those don't modify or cache the content, only serve it to the user which the publisher approved serving it to, and increasingly with HTTPS they can't even see it. Moreover, the publisher is implicitly accepting that by being on the Internet.
Not by network hardware, which is the context of this discussion.
It's also not the type of activity being discussed, which is unauthorized caching: a content provider who uses a CDN is doing so intentionally and while shared local proxies are increasingly uncommon they also respect the Cache-Control headers set by the source — see e.g. https://redbot.org/?uri=https%3A%2F%2Fwww.wsj.com — so again there's the distinguishing factor of authorization.