Google built a bunch of ML systems that work by default for everyone. Personally, I just have a bunch of filters setup that do the same thing. Has taken minimal effort but some discipline to file incoming emails based on my own patterns.
(Note that FastMail does have filters that classify your inbox like Gmail: Personal, Notifications, and Mailing Lists. But while I highly recommend FastMail, I don't use those features because the aforementioned manual sorting rules are better, I brought them over when I migrated as well.)
Here is what I now do instead:
1. I archive all incoming email unless I still need to act on it. Ideally my inbox should always be empty.
2. I unsubscribe from all unwanted email like newsletters and most social network updates.
3. I learned gmail keyboard shortcuts to clear my inbox quickly.
4. I use filters for some mailing lists.
Since I started doing this I am much more on top of my emails. At least for me, most emails in those tabs was stuff I wasn't going to read anyway.
I only had trouble with Google's "intelligent" sorting; you're better off with a proper filter system, like sieve, and setting it up for yourself. Eg. I remember reservation confirmations going into "promotions".