Not a guru, but I somehow stumbled through setting up Thunderbird w/4 different e-mails.
I have used it for years, on various new computers without any problems. Very similar to Outlook, maybe more like Outlook Express. But its a standalone client, has a Calendar for scheduling, Contacts, Encryption, RSS News feed & other stuff. I just use the e-mail (now down to (3)) & the Calendar.
The e-mail I gave up was my own domain so it handles multiple domains. Three others still active are SMTP gmail accts. I think if you want it on cross platforms ("puter, phone, etc) you want the imap option as opposed to POP3. Yep, all the e-mail accounts are in the same client program with their own hierarchy: Inbox|Drafts|Sent, etc. You can set up Local folders in it as well. And migrate too, from other clients, don’t know about from web apps.
Their support page that has very easy to follow setup instructions… hey, I did it! Try starting here: (obviously d/l the program, also on this page). Come back if/when you stumble on some specific issue. Their may be other clients out there but I haven’t looked in a long time.
https://support.mozilla.org/en-US/products/thunderbird
Good luck.