As far as I can tell, there is no software that integrates multiple chat protocols and email. However, if you’re okay with having a separate email application, you may want to have a look at a multi-protocol messaging client like Pidgin or whatever it is that KDE uses these days (used to be Kopete and Telepathy).
EDIT: Scratch that. Kopete and Telepathy don’t have Matrix support. Pidgin still has it, though.
As far as I can tell, there is no software that integrates multiple chat protocols and email. However, if you’re okay with having a separate email application, you may want to have a look at a multi-protocol messaging client like Pidgin or whatever it is that KDE uses these days (used to be Kopete and Telepathy).
EDIT: Scratch that. Kopete and Telepathy don’t have Matrix support. Pidgin still has it, though.
Your operating system is software, and it seamlessly integrates everything you’ve mentioned! Sorry, I’m in a weird mood…
try ferdi, https://getferdi.com/
Thunderbird can do many of them
matrix bridges if you’re hosting your own homeserver.
Doable with XMPP, but a bit tricky to self host. The Jmp.chat / cheogram service does some of that though.
Your browser
You can bridge all of those services to Matrix.
https://matrix.org/bridges/