When I first got my Model 3 it did not support lane change on city streets and now it does, so clearly Tesla does intend for us to use it. That said, I agree it’s really not the best on city streets and you have to pay even more attention to it.
I had a similar situation few nights ago. On a surface steet and with AP engaged, I was approaching a red light. I got that screen with flashing red streaing wheel amd loud warnign nise so I panicked and stepped on the brake. Now knowing that it can stop on its own, I'll give it enough time to respond and hopefully stop.
Anytime you see that screen ("take over immediately") in any scenario, you should no longer assume that Autopilot will assist you in any way. That warning is basically the car saying, "I cannot handle this situation. You're in charge now." So I think you reacted just as you should have.