Real-time decentralisation is the headline. Matrix allows you to host your own server, federate messages in real-time between servers, and still benefit from end-to-end encryption between multiple client devices.
That isn't true at all - Matrix has a rich and growing ecosystem of chat clients. For Slack clones, you're well stocked, but the links given to you by the parent shows there is much more out there.
Thanks for pointing this out - the website is updated now to include poljar's work. (We did feature it on the blog but important to keep the projects list updated too!)
There are several competing Python libraries, those you mentioned plus https://github.com/tulir/mautrix-python. We'll do a round-up and review of the options soon.
There is support in Mozilla Thunderbird - but as a chat mode, rather than email as you might be expecting. You can enable it by changing the value of "chat.prpls.prpl-matrix.disable" to false in the advanced config editor. Still, since it really just enables a chat client, another Matrix client might be better suited.
Good place to get started reading: https://matrix.org/docs/guides/introduction