A gentleman still does in these days.
Yet I was only six years in the navy but I feel vaguely uncomfortable with a hat or cap on indoors. (Glances at avatar) That explains it.
D&R
Of all the folk who have posted, I don’t think anyone has said they are unaware of the convention of removing your hat at table. So that raises the question of why someone would choose NOT to follow the convention. Not sure I can come up w/ any explanation other than a variation of “Look at me! I’m so special!”
Several possible explanations have already been offered. Of course you can dismiss or ignore them as you like, but they weren’t all simply “look at me I’m special”.
ANYBODY who wears a fedora ANYWHERE today is signaling, “Look at me! I’m so special!”
I’m sure is a mix of thinking the convention is outdated and irrelevant to current norms and lack of somewhere to put the hat and a desire to wear one (for whatever reason–I’m not a hat wearer). The guy put on a hat that day for a reason, and wore it at the restaurant for the same reason.
Like I said up-thread, I know I’m not supposed to put my elbows on the table, but I like to do it sometimes, it doesn’t hurt anyone, and I really doubt anyone is offended by my behavior (although I’m sure they think I was raised by monkeys or something)
Hat hair is a perfectly valid reason to not want to take your hat off.
Would you rather embarrass this person just to satisfy your rules of etiquette?
I’ve read, and barring evidence to the contrary I’ll assume it’s true, that the original custom was to take your helmet off inside to show that you meant no harm to others and recognized that they meant no harm to you. Somehow, this custom spread to folks who were wearing hats and not helms and persists to this day.
Re Following Social Norms
I think that a whole thread could be devoted just to this subject.
Growing up, the running joke when one of us committed a faux pas was that meant that we were unlikely to be invited to Buckingham Palace/the White House for dinner.
The theory I’ve heard is that filthy hats among all other clothing is uniquely positioned to drop filth into your food while you eat. If you come out of the rain and don’t remove your hat it’s about as courteous as dragging an open, wet umbrella around someone’s house or business.
8 years USAF 30-mumble years ago.
My only hat-wearing now is a team baseball cap to outdoor baseball games. I still take it off when in the innards of the stadium; to do otherwise is unspeakably wrong/uncomfortable. And I tuck the bill into the back of my pants/belt to store it just as we always did.
Some changes are forever.
It’s one of those double standards in which women can do but men can’t for spurious reasons. It is a fashion statement for men as well as for women. No one should question a man with a bald spot who prefers to cover his noggin with a fedora. Which brings up a pet peace of mine, women who place their purses or pocketbooks on the dining table. 
Just kidding! Or am I?
I wasn’t aware of the tradition that men had to remove their hats when inside, excepting church, before this thread. It’s not something that’s ever come in my personal experience. I’m 57.
I can see why women would have a different etiquette. Of their hats were placed in specific ways and held on with a long pin. Putting it back on would require a mirror and could take some time. A man just has to plop in back on.
Which are another way of dividing societies. In Switzerland, if I were to eat with my hand in my lap, people would wonder what I am hiding.
Etiquette based on gender is outdated.
Simple and elegant. Why worry about everything else?
As long as fashion differs between genders, the etiquette of fashion will differ and at least some of that will have a legit basis in logic.
But otherwise I agree 100%. e.g. Holding doors open for others following close behind is polite. Doing it for women because they are women is outmoded. As is not doing it for men because they are men.
And please, please, can you just walk into the elevator and get out of the way instead of standing IN THE ELEVATOR DOORWAY “politely” waiting for me to enter first?
(Maybe I should start my first pit thread.)
I wave them inward/outward with a smile and if they stand there obstructing traffic I just walk into those people as I enter, my shoulder not quite clearing them. Oops; I’m clumsy. Big smile. Kill 'em with kindness while delivering a little life lesson.
Gave it up for COVID, but it’s about time to restart that habit.
Elevator etiquette is misunderstood and clumsily applied. The really old-school etiquette was that a gentleman enters first to guide a lady into a moving conveyance, such as an elevator (“ladies first” was never correct for elevators). The sensible thing to do is whoever is closest to the door goes through first, otherwise it creates a jam like a scene from the Three Stooges. I still see men fall all over themselves to let women exit first when it just clogs the flow up for everyone, and isn’t really that much of a favor to the women who have to run a gauntlet of men to get out.
As for “ladies first,” I agree that gender distinction is not what it was in the Victorian era and modern behavior should be whatever is reasonable, putting everyone on an equal footing.
Once I had been waiting for my turn at a deli counter and a woman walked up right in front of me and started to order. I said, “Excuse, I was waiting first.” She said, “What happened to ‘ladies first’?” Well, that is some bullshit right there.
The response to that is to reply “When I see a lady, I’ll let her go first.”
Not only is it not much of a favor, it is actively unpleasant if i need to squeeze past them. And I’ve seen men on-line say they like to do it just SO the hot young woman will get close to them. That’s just creepy. I’m no longer either young or hot, so i assume men doing this think they are being polite, and not just hoping I’ll brush into them. But it’s still really annoying.