It's not only unethical and illegal, it is easily proven to be both. Law enforcement in the United States has, as its ultimate authority, a foundational legal document known as the Constitution. This document makes it very clear that it is illegal to search someone's (not even citizen, mind you, this applies to all humans) person, papers, or effects without a warrant.
The only reason to "scrape social media" is when you're doing it without a warrant. If you have a warrant, it becomes quite easy to request that data directly from Facebook.
> People seem to think social media is akin to private communications where it's more akin to the public square.
In any other argument, I would agree that it is a "public square". But the police specifically aren't permitted into this public square when acting as police officers. It is unethical and illegal.
> While you have a constitutional right to not be searched without consent/probably cause,
No, not even those things are sufficient. You have a right to not be searched without a properly issued search warrant.
> you do not have a constitutional right to spouting off in the public square without consequence.
In fact, you do actually have this right.
> Putting out an IG post of yourself with illegal guns
You cannot post on IG with "illegal guns". The only thing IG allows you to post are pixels in raster images. Photoshopping yourself holding the BFG-9000 isn't a crime, and it's not even "probable cause".