I've already read that thread. And in my experience all your assumptions are false as far as I can tell. I have Aggressive Doze and Doze on the go active, and still receive IMs via apps set to ignore Doze on time.
You have not read the other thread carefully, then, because the point was about emails, not IMs. IMs work with Doze? Yes, of course. Google has decided that IMs are higher priority than emails, and has structured its GCM accordingly. Their official documentation mentions as much:
https://developers.google.com/cloud-messaging/concept-options#setting-the-priority-of-a-message
Why it has decided that it should be so, only they know, and why users have no option to change this is beyond me, since that's an idiotic, dictatorial Apple-like my-way-or-the-highway approach. For example, there are people for whom instant push email notifications are more important than instant IMs.
If your phone isn't exempting apps from Doze properly, I'd expect it more a problem on the app itself, as you also noted on that previous post.
The point is not that a phone doesn't exempt apps from Doze properly; the point is that Google has conceived Doze so that very few things, namely high-priority GCM notifications, phone calls and alarms can wake a phone from Doze; Google has specifically conceived Doze so that a whitelisted email client CANNOT wake the phone from Doze to check for emails, i.e. Doze breaks push email - unless push is via a server implementation, as opposed to, say, IMAP idle, which relies on GCM and which sends high-priority notifications. This is mentioned very clearly in Google's official documentation, which I had linked in my other thread.
You mention that gmail and inbox work correctly. Could you please elaborate on this? Last time I checked, neither sends high-priority GCM notifications. Has this changed? Is there a way to make push email work correctly with Doze? If you know of such a way, any colour would be most appreciated as I'm very interested in the topic. Have you specifically verified that gmail and inbox can receive emails while the phone is in doze? If you have, that's the first I hear of it, so, like I said, I am most interested in any colour you may have; in fact, the web is full of complaints of gmail notifications not working properly while in doze. Are you sure that it wasn't something else (e.g. a whatsapp message or a clock alarm going off) that caused the phone to wake from doze?
In fact, Google's official documentation mentions that google will not grant exemptions from Doze's (flawed and idiotic) power saving feature for email apps which do not use GCM:
https://developer.android.com/training/monitoring-device-state/doze-standby.html
Based on your own research, Aquamail doesn't send high priority notifications. Why? That's not a problem of Doze. That's an app developer's problem. If other devs can do it, why not them? It's a matter of getting the devs to code the notifications properly, not an issue with Doze itself. Google gave them the way how to work with it. It's up to the devs to do it properly.
This comments suggests you have some confusion on the different systems of push email. K-9 mail and Aquamail rely on imap idle; Google has decided that imap idle is not to work under doze, so the only email clients which can have functional push email notifications when in doze are those with a centralised server-side implementation which relies on high-priority GCM notifications. In other words: with imap idle, the client connects to the server and tries to keep the connection alive listening to new emails. With GCM, you need a centralised server which connects to your email server, listens to new emails, and sends a GCM notification to your phone when a new email arrives. This is almost the same concept as that of a Blackberry server.
So, no, it is absolutely not correct to say that Goggle gave them [the developers] a way to work with it.
Apple fanboys like to repeat that iOS doesn't support imap idle because of the battery drain, but in my 6 years of using imap idle on android on multiple devices I have never experienced battery drain; in fact, I have always found the difference in battery use with imap idle on vs off absolutely negligible. I appreciate that a server-side implementation may drain even less battery, but I don't like this solution because it means giving a third party, other than my email provider, access to my inbox, and I don't trust privacy policies and marketing guff claiming that they don't look at my emails.
Finally, as of now, AFAIK no email client exists which sends high-priority GCM notifications. Why that is I don't know; however, setting up a server which listens to your emails and sends GCM notifications (regardless of priority) to your phone has a cost (who maintains and pays for this server?) whereas a system like imap idle, which does not need a separate server to check your emails, does not.
