If I ever get a notification on my phone that isn't a human trying to interact with me I immediately disable the notification and/or app.
With email I often (daily) unsubscribe from whatever impersonal newsletter shows up in my inbox.
I save my attention for people who take time to message me personally. Red badges on my phone are irritating.
Whitelist: Inbox. High-priority senders.
Blacklist: Trash / treat as spam. One model is to add an address to your blacklist as you unsubscribe it. Any further messages feed spam filters.
Greylist: Previously unseen. This goes to a bucket for sorting through. You can apply more heuristics to this (keyword / sender / references, etc.).
Backlist: Known low-priority senders. It basically goes into the "deal with it later" file.
Anything that's specifically sortable (e.g., technical mailing lists) goes to its own space.
My fundamental problem is that few tools offer the means to do this type of sorting.
Gmail's filters are, the more I use them, increasingly pathetic.
Procmail is powerful -- I've built up highly extensive filters with it over decades. But it's also a pain to use, largely unmaintained and unmaintainable. It lacks a simple means to, say, point at a list of addresses and say "good, bad, or ugly". Adding addresses specifically to rulesets is somewhat tedious. And as with other cleint-side tools, it doesn't work with IMAP services in a reasonable way (you can treat IMAP as POP, but then you lose filtering for all other clients accessing the server).
There are some other tools -- there's a ... Perl? based tool, Mail<mumble>, which is cleaner than procmail but also different and I've not wrapped my head around it. There are IMAP-based tools, but I've had little luck with them.
Email's generally simply getting crufty. I've resisted calling for a replacement for a while, but increasingly, it's time.