It's not like you're being force-fed notifications against your will. And it's not like offline content hurts you. Any inconvenience offline support causes to you pales in comparison to how much people benefit who actually need to be able to access content on spotty connections.
But for notifications, even the only-once* prompt is an annoyance we never had to deal with before. It's like the European Cookies Law: great in essence, but when every website bugs you about it then the problem has been exacerbated, not solved. I don't care about receiving notifications for a website I visit once in a blue moon, and I especially don't care to be even asked. There should be an action that triggers the prompt, like some kind of opt-in.
Edit: same with "app banners"
*it never is, because cross platform, multiple devices, multiple browsers. A fucking pain.
My thing is: I visit a particular website to get the latest news on a topic but it's done with some kind of poorly coded implementation of offline cache. Despite my privileged, first-world, 4G-everywhere connection, it insists on loading stale data even though it looks exactly like a regular website. And this clashes with my vision of internet, which is, as others said, regular pages with hyperlinks, and a Refresh refreshes the page to get the newest version, even if it's stale. If things haven't changed, then they haven't and I instantly know nothing's new.
As I've said, pretty minor and I realize that Offline mode has much more pros than cons.