From what I understand, IMAP IDLE on a smaller battery powered device has some drawbacks. Again, I don't know the details of the ins and outs when it comes to Android and mail push there, but just throwing in IDLE support on its own basically isn't as simple as just doing in and that's it.
IMAP IDLE is rather unsophisticated. It requires one persistent connection per mailbox and doesn’t allow the client to distinguish between the signals that the server sends. If you have lots of folders and are doing a lot on your email account, such as deleting and moving emails around, IMAP IDLE is overkill because it will attempt to reflect every action you take on your phone as well, even when you are not using your phone. The command was never meant to minimise battery impact, but make your email account more efficient without having to force your mail client to check sporadically whether you’ve made any changes on another device. In other words, good for desktop computers, not so good for mobile devices.
Exchange ActiveSync was specifically designed for this and requires only one connection per account, not per folder. It also synchronises the changes more efficiently. That’s probably why Apple doesn’t use IMAP IDLE even for iCloud, but a different implementation altogether. There are better ways to do this, like P-IMAP and IMAP NOTIFY, but I don’t think they are widely supported yet anyway.
With respect to other devices, I remember that my BlackBerry 10 device was not great with IMAP IDLE either. By default, it only used IMAP IDLE on my inbox and sent mailbox, all the others were disabled and merely refreshed by regular fetching. It was a very weird experience, because any changes outside these folders were never pushed back to the desktop or web client, but only submitted by fetch. When I would move an email to another folder, the email virtually disappears from the inbox on my other device and only reappears after the phone has updated the receiving folder again. It’s really not a good experience and when you select push for all other folders, you will notice it in your battery life. It’s just a waste of energy.
With Exchange ActiveSync you often run into the problem that providers cap the sync limit to one or a few months and they don’t sync anything else. My own mail provider offers ActiveSync, but I can only ever sync all emails I received within the last 30 days. So even if you get push support, ActiveSync has other drawbacks. I have become accustomed to not having push email support and use a 1 hour fetch now. For that I get access to all my emails.
The Gmail app can offer things that the Mail app cannot. Google can use Apple’s APIs for push notifications whenever the user needs to be notified (which doesn’t require much energy, since its collated and goes via Apple’s push servers) and use background app refresh and push triggers to sporadically look for changes when it actually matters. It’s probably a much better experience to begin with.