Then you also need to build and distribute Android and iOS apps for your employees. Congrats, you now have a significantly larger project than "put Mattermost on a server and update it occasionally".
I am using it on both cloud-hosted (work) and self-hosted (friends chat) servers.
Push notifications work fine, the mattermost app that's already on the android and iOS stores work fine, they have done for the 3-4 years I've been using it.