FastMail is using JMAP, a protocol designed to support push notifications. MXRoute is stuck with SMTP/IMAP which don't support those. There might be a different reason, but I think this is probably the core issue here
The stock iOS Mail app does not support JMAP and therefore it has no relation to Push notifications for the stock iOS Mail app.
I think saying they are stuck with standard email protocols is a bit of a stretch. JMAP is not widely implemented outside of Fastmail and certainly isn’t used by Apple Mail, which actually uses a proprietary IMAP extension (XAPPLEPUSHSERVICE).
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For example a relatively modern email protocol: https://www.rfc-editor.org/rfc/rfc8621.html
FastMail got access to it and it is not clear how, without any public announcement or documentation.
It is as if Apple just picked a favorite and went with it.