I don't know, by the same logic you could say you have a right to not let other people look at you in public (you don't). I understand the desire for a right to privacy (despite no such explicit right being anywhere in the constitution), but when you are in public, you be definition give up some privacy.

The thing is, I think most people who object to this so much would probably never even visit those bars where people care what the "scene" is like anyway

This. I'm having serious difficulty imagining that anyone would object this much if the bar simply hired someone to sit in the corner, count the men and women, and post the numbers on a twitter feed.

Because it's a very short hop from what's described here to full-on tracking, and some marketing company knowing where you are, what you're doing and who you're with at all times.

It's a similar situation to Google's street view - sure, it's in theory in the public domain, but when you start aggregating this sort of thing and making it searchable, it's different.

a bar is not a public place, it's a private place that the public are invited into.

Aren't bars more formally referred to as "public houses"?

sure, but public in "public house" (or just "pub") means open to the public, it doesn't mean that the public have some sort of right to access, whereas a public street means it's accessible by everyone.

Relating to the usage of the word public, humorously in England we have "public school" which is the American "private school", in America if you go to public school you're considered normal and in England if you go to public school you're considered rich.