Assign by hand from the card
Each request gets a card in your triage channel. Choose Assign to me to take it, or Assign… to pick a teammate under Owner. The card updates in place, so the whole team sees the new owner without another message in the channel.
In the web app, the owner menu on a request does the same, and Assign to me is one click. You can pick only your own teammates, never people from the customer's workspace.
Can someone take a look? It's blocking our release.