I had a very similar experience in the supermarket a few months back. A man coming my way whilst shopping, I look up, he looks at me, and WOW! Recognition, chemistry. I almost blurted "Hey, how good to see you again!" but within a split second realized I didn't know him at all. Yet, I do know him. We both smiled, then moved on.
However, at the cash register I noticed I was standing behind him. He noticed me too and smiled and said goodbye before he walked out the shop.
Had he asked me something, maybe leading to a cup of coffee, I would've stayed for sure. Not because he was my type, he wasn't even overly handsome, but there was this weird connection.

And yes, leaves you wondering, what if?

I've had both good, and bad experiences
from being approached by random strangers
in public places.

I simply prefer, that it NOT happen...
At least, not by anyone who lives within
a 50 mile radius of me.

Yea so true. My mind starts working again about 2 minutes later. And then I can think of loads of things I could have said.
Fortunately my mouth continues working without the mind taking part.

I'll say something silly like "come back" a little forlornly. after the initial freeze the back up brain cell kicks in. I know it's going to be a couple of minutes until the other brain cell comes back on line.
Then I'm back to full power :)
Have faith..

"What if"... My two favorite words when I want to spend minutes, hours, days, and sometimes weeks pondering what could have been or what may be... the merry-go-round of living life in the past or future rather than living in the moment, the present.

Haha, been there many of times. It can be quite a challenge to not engage in the "what if's" of life. I'm amazed every time I choose not to get on that merry-go-round and simply act on what I'm wondering about. To simply live in today.

Ouch.

I'm thinking about the guy in the OP scenario.

Smiles at a girl, she smiles back, he goes to approach, and she runs away.

There's a blow to the ego.

...practice really does lead to forms of Perfection for men.

As she turns and walks away my head hangs a little lower. Is it just me I wonder? It's been so long since i did the whole dating thing, I think to myself. It's harder than I remember. One of these days though I know that special someone will feel the same spark. Funny thing though, just for a minute I thought this might be the one.

The end of the day for me I wouldn't look twice cause he was a bloke I would have just been looking at you and I would come straight over to you x

Most of the time we have one short time period to make that initial contact. When we don't then it is all about "I wonder" and we never will know!! Grab life, you only have a fleeting chance!