Regarding Picard's actions and views in "Who Watches The Watchers"...
I am in complete agreement with Picard's actions and reasons. If the Minkatans were to go down the path of making Picard a god, that could very easily be subverted into inquisitions, crusades, holy wars, etc... exactly the things that have happened in our own history. And still go on in some parts of the world.
Picard very correctly did his best to make sure he wasn't deified. I don't see how his actions and reasons can be construed as anything but trying to fix a problem.
About the topic question...
It's an interesting question. I don't think he would be very much like the Picard we all know. First, he lives in a post-scarcity world. Second, a lot of the options he would have had growing up don't exist now.
So I would have to look at his main interests... exploration and archaeology. Our ability to explore is somewhat limited, but there is a lot of archaeological opportunities available now. (At least, in comparison to being, say, an astronaut. There are a lot less missions going to space than there are digs unearthing clues of the past.) And archaeology is exploration... it's simply exploring the past.
So I think Picard would be a full blown archaeologist, particularly his dialogue from "The Chase" that says if Starfleet wasn't an option he would have done that.
On a side note: if Chakotay was alive in 2023, he would likely work in the same circles as Picard because Chakotay said paleontology was his first love. Both professions do a lot of digging.
(For some reason, I have an image of the Atari game DIG DUG in my head with Picard's face superimposed over the game's hero.)