So I put attention on this and yes my S22 is doing the delayed Gmail notifications also if the phone has been idle for a while. I just picked up the phone and there were some camera alerts waiting so those got through. But there were no Gmail notifications until I unlocked the phone, then immediately the Gmail notifications of several emails came through. This is what used to happen on my Pixel also and running the batch file fixed it until the next boot. I am not sure I will bother running it on this phone though as these days I don't get emails that are so important that they can't wait until the next time I happen to open my phone. But if I was still depending on getting notification of work emails as soon as the email arrived, I would run the batch file.
I plan to try out FairEmail so I will see what happens with notifications with that.