Except you have no idea what else they're doing in there before they so nicely "close the door".

Besides, your metaphor doesn't work since cops aren't allowed to just enter a premise due to it being unlocked or open. That doesn't fall under any of the reasons to enter a premise without warrant. Leaving a door unlocked or open is not a crime that gives them free pass.

I really have a hard time understanding people's logic on here sometimes. Apparently the entire US government is systemically racist and evil but these same people want to give more power to this government? Which is it?

Really? They can't set one foot inside in order to reach the doorhandle if the door is 90 degrees open?

And what America are we talking about, "your rights according to the ACLU" America, media portrayals of America, or in-real-life common sense America where you can have a ten minute conversation with a policeman about everyday things with no worries because it's actually no big deal?

Yes really and this isn't just simply "door closing".

https://www.law.cornell.edu/constitution/fourth_amendment

I guess. Reminds me of the time in Santiago de Chile we were fixing up a place and the door was ajar. A group of Jehova's Witnesses came in without knocking, holding their pamphlet in front of them in a noticeably odd way, in case they found anybody inside they'd be all about the pamphlet. I guess they thought it was their warrant.