Why doesn't Google's auto-sync work on Android?

Question

Android

Hello.
Google account is not auto-syncing. This is manifested in the fact that mail does not come to Gmail right away or does not come at all. Until you manually update in the application itself or in accounts, mail does not come. I tried reinstalling Gmail, reconnecting my account - it does not help.
There used to be similar tops with whatsapp, pushbullet, etc., but it went away, it’s not clear how.
All checkboxes, user settings of applications are rechecked by 10 times.
In which direction to dig? Maybe there are some programs for setting up / monitoring synchronization or push notifications for applications or something like that?
Android 5.0 rooted stock for Asus zenfone 2 (WW-ASUS_Z00A_2.19.40.18)
PS Please do not offer a flashing / Hard reset, there is not a single custom firmware for this phone, there is no recovery to save the data, so I want to do without it.
Update: recovery has already appeared, but still I want to do without flashing.

Answer 1

If the synchronization button is disabled, letters will not come, even if there are checkmarks everywhere.
Please note that with active synchronization, the battery will drain faster.

Answer 2

The sync button is on, of course.