If you go out to eat at a restaurant or are out shopping do you just leave your tiny LO's?

Do you just keep them in the car seat/stroller the entire time?

My baby doesn't have awesome neck control yet (or she is good at sitting but tips over a lot) so when we go out I would always keep her in her car seat. However since I am working on correcting her flat head I need tips. I dont just want to keep her in the car seat!
we go out and about a lot.

The last time I wore her in my moby wrap at costco facing forward and she loved it, but I dont think I could do that at a restaurant!?