Nothing, but there already exist many other Matrix clients (shitty as they may be at present), as well as (IIUC) an Element PWA that uses web push (which is IIUC supported by Synapse) for notifications. Synapse also (IIUC) can be configured to use an arbitrary push gateway.

This is what I mean by "generic" in the other comment you replied to. I appreciate the value of tightly integrated server and client applications, and fully believe that Zulip's implementation of notifications may be both a) better for usability and b) a lower maintenance burden for the development team than supporting web push in a PWA, but---again---I am looking at this from a certain perspective where the way Matrix is architected and the breadth of the ecosystem imply less long term risk for my use case.