The "network connection" provision of AGPL doesn't require you to open source anything you use to talk to it, just modifications of it. To make it clearer:
Client <--> Server.
The server is AGPL licenced.
I can write a client to talk to the server and I don't have any licence restrictions. For example - if I write a web client to talk to an AGPL web server, I don't have to open source the client.
If I modify the AGPL server, and then put that modified server on the internet then the AGPL terms require me to make available the modified source code.
> Our desktop client is closed source, but you can use Element (or any open source Matrix client) if you prefer.
I see most bridges are licensed AGPLv3 [0]: Aren't you required to AGPLv3 the desktop client, too?
[0] For ex: https://gitlab.com/nova/whatsapp/-/blob/master/LICENSE