They're an important safety device in our culture where one adult is often in charge of multiple children.
I had my eldest on reins last summer. I had my newborn and just turned 2 year old in the double pram and my eldest who was 3.5 years old on the buggy board. We were on a long day out with friends near a lake and a busy road, I couldn't risk him deciding he was going to get off the buggy board and make a break for it as I would have had to leave my other two in the pram to run after him.
I didn't use them the whole day, just for the high risk parts of the day and I took them off at appropriate times where I judged it safe for him to explore so he could play with his friends. He also understood that if he didn't listen to me and tried to run off at any point that I would put the reins back on. He was completely unbothered by wearing them and understood why they were needed at that age too.
The irony was I had three children to handle solo and the friends we went with had just one child each the same age as my eldest. But they were the ones struggling as their children kept running off and they had no way of stopping them. We might think they're "grown up" at 3.5 but they are still little, have a limited grasp on the dangers around them or to think logically. So they need us to act to keep them safe and sometimes that means reins.
Nobody has ever made a comment to me about them and if they did they would wish they hadn't! I do however get plenty of comments along the lines of "you've got your hands full", I certainly do so I make sure I can keep them all safe.
I also always keep a pair in the back of the car for unforeseen situations where I might need them and in case the car were to break down and I needed to keep multiple children from running into a road while we waited for breakdown assistance.
How many of us wore reins as toddlers and are traumatised from it, I would guess we are all just fine and importantly alive and well!
Ignore that woman, she doesn't live your life. Safety first, always.