Shortly after the time that the original author/maintainer of Phabricator announced his intention to stop maintaining it, the group of people who got together with an aim to continue/take over development and support used a service called Zulip to discuss how they'd do that. I remember feeling fairly impressed by the service at the time, as it seemed an improvement above my experiences with IRC, Gitter, and Matrix.
It meets most of the criteria established at the top:
TLS encryption
between client and server; and all data is encrypted at rest
be open source
Source available on GitHub: https://github.com/zulip/zulip
be libre,
Apache 2.0 license
be free,
As an OSS project we'd be eligible for their "Standard" cloud package for free
ideally not be self hosted or at least part of something we already host.
Is cloud hosted (with self-hosting as an option)
provide a web chat, for people not willing to install extra software
A web client is available (as are desktop, mobile, and terminal clients)
Tagging specific channels as public is possible, so no need to sign in to read chat
should allow logging the channels for posterity
Ability to export select (e.g. publicly viewable) channels into a static HTML archive, with incremental updates where possible - https://github.com/zulip/zulip-archive
There's also bridges for IRC/matrix/XMPP/etc.; optional voice/video chat integration (Jitsi by default); private channels/topics; user administration & moderation; etc.
Of particular interest to us, the service supports what it calls "linkifier"s - detecting phrases such as "D1234" and adding a hyperlink pointing to the relevant issue/commit/whatever. (How this appears in an exported archive I'm not sure.)
And it supports integrations with gitea, trac, phabricator, transifex, jenkins.
Anyway... just suggesting it as an alternative to what's been mentioned thus far. Thoughts?