At least on iOS, you can go into the system settings, scroll down to the Messenger app, and untoggle "Allow Notifications".
It's having to go around Messenger, and doesn't disprove your point at all. But I ran into the same realization you had, but couldn't easily stop using it completely as Messenger video calls are really important to my grandparents. So I just disabled Notifications at a system level, and open it up periodically to check new messages (same as you mentioned for email/text).
As an iPhone and gmail user, I used to think that not having gmail push notifications in the stock mail app was a drag. But now I realize that it is actually a blessing in disguise. Sure, one can manually set the mail app to check for new mail via a "fetch" schedule (or not at all), but something about there being no option for true push is quite nice.
* Read part of a book before starting work.
* No HN/reddit/etc until noon.
* No Facebook until 6PM.
* Disable almost all notifications.
* Only check emails/texts once an hour at the top of the hour.
* Use software that respects my desire for focus.
The last one is currently the most interesting to me. When I was going through disabling all my notifications, I realized you can't truly turn them off on FB messenger. You can only mute them for up to 24 hours. After that I decided to remove any software from my life that was created to manipulate users rather than empower them. That's mostly meant switching to OSS apps and self hosting.
https://github.com/Kickball/awesome-selfhosted has been an incredible resource