Thanks for this bug report and for the helpful screencast, @AaronChaos.
This seems specific to Firefox (including Firefox Developer Edition). I can only reproduce it there. It seems the way Local generates certificates doesn’t meet Firefox’s expectations for self-signed certs.
I have logged this as a bug for the Local team to address in a future update. For now SSL should work for you in Chrome or Safari, but I’m sorry that it isn’t working by default in your preferred browser.