It sounds like the /etc/hosts file may not be getting updated. Curious, do you see a warning banner at the top with a “Fix It” button?
One way you can work around this is to switch Local into the localhost routing mode by going to Preferences » Advanced and changing the routing mode to localhost. For more details, see https://localwp.com/help-docs/router-mode/
Update, I found Localhost Routing in the preferences and switched to that. It got hung up on “provisioning” for a long time so I just quit Local and restarted it. Now it works as intended. Thanks for your help!
Also, now that I think about it, if ERR_CONNECTION_REFUSED is the error, it sounds like Local’s routing layer isn’t able to start. The quickest way around that is to switch to localhost routing as you did.