I suspect that the cost for push notifications might have to do with the costs imposed by Apple and Google for their respective OSes' notifications backends, but don't quote me on that. The $3.5/user/month smells like a way to recoup that kind of cost without making it overly burdensome for folks within one standard deviation of the mean.

That said, it appears that XMPP might be workable here: eJabberd is still actively maintained, and there are mobile clients (namely, Monal for iOS and Conversations for Android) that appear to support push notifications. Given your talk of self-hosting, I feel comfortable recommending something that requires some degree of system administration investment.