I went looking once at my state's vehicle code, which was at once exasperating and quite illuminating.
There was a part about how you cannot modify the vehicle's body in any way, which includes cutting it, welding on it, and looking at it funny. Presumably this is for legal and insurance purposes so that if you get into a wreck and your chopped or tubbed lowrider or hotrod falls apart, you can be blamed for it.
But there was another part about how removable parts of the body can be removed, presumably in reference to hardtops, say. It did NOT specify the method of removal, so anything that bolts/unbolts would seem to be fair game. The law isn't written all at once, but over time, so some parts will conflict with others, and the government would only "fix" it by adding MORE rules
Point being that you won't find a single, straight answer, save this: If the cop says it's unsafe, it's unsafe. Any state's vehicle code will have a clause that says if, in the opinion of an officer, it's unsafe, it's illegal. Period, end of story. The *maximum* speed limit may be e.g. 55MPH, but if it's dark and foggy or whatever, the cop says in his opinion it's unsafe to do over 35 ... it's a 35 zone.
So, if you drive wildly, weaving across multiple lanes without using your signals and shouting profanities and passersby -- or even just squeal your tires leaving the bar's parking lot right in front of the cop -- don't be surprised to get a ticket about your doors. And your tail lights. And your seatbelt. And anything else the officer feels like, if you piss them off.
If you drive reasonably and respectfully, I'm sure you'll do fine. I've had my topless, tailgate-less, half-door, no-windows-but-the-windshield Blazer out on the street many a time and had no issue. (Funny looks from said passersby, but no trouble with the law.)
I did get stopped once in the other truck for a seatbelt check; had my lap belt on, but the officer was apparently expecting shoulder belts. After politely explaining that it was a 1974, implying without making a big deal out of it that shoulder belts just
weren't back then =)), he sent me on my way. (And me without my license, as I was literally a block from my house, just doing a quickie test drive. I offered to walk back and get my wallet, but he just groused at me to never do it again... prolly pissed 'cuz he didn't get the ticket for his quota

)
As you say, there's lots of Jeeps (and Scouts, and early 'bobtail' Broncos) that came sans doors; so unless you piss the cop off, why is (s)he gonna try and figure out what you're driving and what it should or shouldn't have had decades ago? Being polite and respectful trumps the letter of the law.
-- A