When I tap notifications, it does say "on" and the little slider it set to the right, which also indicates "on." I don't see anything like what Rukbat described, though -- a setting in which I can chose HOW OFTEN it checks.
The way phones are today they try limit as much process going on in background and focus at battery saving , so you'll have to try turn adaptive battery in battery saving mode and some apps like we discussed earlier , looks like they either want great battery life at expense of having immediate notifications or slightly worse battery for immediate notifications by disabling things that optimize battery.The only thing I had to change was in the second screen, in which selected All instead of 30 Days.
For all of this discussion, though, my concern is that this is just a general issue that can't be fixed by you or I. For example, I came across this article today: https://www.androidauthority.com/delayed-gmail-notifications-1052389/ . If so, that's really bad and something that needs to be fixed ASAP. This article is from November, so that's already too long as far as I'm concerned. For all the things they can do, a smartphone is, to me at least, still first and foremost a communications device!