I think Blackberry had the right solution with their Hub app. It was just a big inbox with a lot of integrations. It would show you all the conversations from different apps (including email) with all the typical filtering, sorting and searching. You had a single screen to see all your conversations nicely organized but clicking on a message would take you to the external app to send messages. It was super useful but of course apps had to provide some API for this to work and BB wasn't popular enough so at some point it just stopped working. Android could easily do this but I think they are just not as good at UX as BB was (seriously, BB OS10 had lots of great features iOS copied a decade later and more, it was really nice).