Google Delta Chat. It’s a chat protocol/server/client based on email. You only need the client - it will use your existing email account (and your recipients’); if you use the Delta client, it will appear as chat; if you don’t it will just be 2 line emails - and similarly for your addresses.
When/if the actual server comes of age, you’ll have additional features like online indicators and stuff. Bit it’s already usable today (if somewhat wasteful in bandwidth and storage compared to e.g. Signal) and there are no new accounts to register or discover — it’s usable with standard email as transport.
I wish this were more popular. I love the idea of using the email protocol for chats. Email servers are ubiquitous and, better, everyone can email everyone else and your contacts and you don't have to be on gmail or yahoo or anything else.
I recommend (a recent) ejabberd if you don't know anything about XMPP: it does an excellent job coming with good default, clear, commented and foolproof config and once it's running you can literally forget about it.
XMPP does manage offline message catch-up across all devices, and with good mobile clients I've never felt the need to redirect to email (especially these days with ubiquitous E2EE, that's not something you would want the server to be capable of doing), that said I believe it would be "easily" scripted up.
