I use Element X every day. On the phone it's very buggy and doesn't get push notification for the latest messages sometimes.
But it has only happened a few times for me. I have not created any github issues. I only have 1 friend on matrix so I don't get that many messages :)
But each time I open the app, the syncing is kinda slow, much slower than on the computer in my experience. Isn't it possible to enable fetching in the background?
That said, from these symptoms, it really sounds like you are on an older build of the app - and probably using the old sliding sync proxy rather than native sync implementation which has now landed in Synapse. The new builds also fetch in the background (whenever you get a push).
Please upgrade, try again, and failing that, file bugs on github.com/element-hq/element-x-android!