When my own kid was very young I was paranoid about touching him in public. It didn't stop me, but I was mindful of it, and I would occasionally feel I was getting extra scrutiny by some parent; a man not at work in the daytime, holding a young boy's hand.

Feh.

This strikes me as odd. As a parent (dad) of two, I wouldn't ever dream of feeling odd about holding either of my kids' hands - or even for that matter chasing them down when they run away screaming (which happens often).

Just a counter data point here.

I didn't feel odd for the act, I felt uncomfortable with the assumptions or speculation implied by the scrutiny. No one likes to be distrusted, regardless of the context.

I wear a shirt that says "Free Hugs", and I've had many adults take advantage of it. The other day, a kid asked me and I got all weird and "can't"... I don't wear that shirt anymore.

Absolutely true. My g/f coos at other people's kids, but I look the other direction. The last thing I want is a liability for friendliness.