There are some push notification addons as well as an app being developed.
Will these make your users happy? Does it replace Tapatalk? There is a pretty good chance the answer is no. The same people you hate for using tapatalk usually don't wish to use a separate app for just your site or just xenforo sites. The uproar wont be as bad as removing tapatalk and not providing anything else. But such a change can do signifigant damage if you don't check your demographic first.
I honestly forget to install the app on my phone anymore so there are quite a few sites including the ones i develop that I rarely read my conversations or new posts mentioning me until several days after the fact lol.