I only had one child, but when she started screaming because she wanted to leave, the very last thing I was going to do was let her have her way.Good for you! Parents are to be parents, not buddies or bestest friends.
One of mine decided to sit and refuse to walk in the mall. We left him where he sat. He couldn't see me duck into a store to watch out for him. When he realized that we weren't going to put up with his tactics he stopped. We also had several grand parent age shoppers who encouraged us to keep doing what we were doing.
Done right, spanking is a last resort and done very sparingly. As you can attest, it can be very effective.
I also want to put in a good word about men in this discussion. My husband backs me up when I have to discipline our son. Since I am the one home with him the most I get the most tantrums etc. I have one friend who when she does finally try to discipline her child her hubby doesn't back her up. Guess he had a step-dad that was a beater and he can't separate the two. It is really sad because he is hurting his daughter in the long run by not teaching her limits etc. This little girl responds to total strangers who take the time to correct her. I mean, she listens. Yet, she can be acting like a total monkey and Daddy just continues with his conversation like she isn't even around.