My Moto X 1st gen started delaying Gmail syncs and associated notifications until I opened the app. This had never been the case in the two years prior with the phone. I was having other “older phone” issues and got a new Moto Z Play and I still have the same issue. I use this for important communications and can’t have it delaying sync indefinitely despite sync settings. Doesn’t matter if I’m on data or wifi either.
Have you checked your on-line Google security settings? Perhaps they are asking you to confirm access to your account from this device.