Can you imagine the cost of ever doing the Last Supper?
It was a passover seder, so I have a pretty good handle on what it would cost to do. 
If there’s an extra seat for me to put my hat on, I put it there. If there isn’t, I wear it.
A hat is OK at a truck stop, as long as it says Peterbilt. Otherwise, no hats on in restaurants, homes, when the national anthem is playing, or elevators when ladies present.
I’ve only been inside a VFW hall to see a punk band back in the Eighties, but don’t the regulars wear those envelope caps when they’re dining/drinking there?
My general view is that there are certain archaic etiquette rules that we can do without. I have no idea why you cant wear a hat at dinner, wear white after (or is it before) labor day, and many others. Just be a nice person to your fellow humans
I’m not a member of the VFW, and I’ve never been at an indoors veterans event, so I don’t know. But the VFW cap isn’t a military cover, and VFW members aren’t active duty military members in uniform, so they aren’t bound by military uniform protocols.
But do they wear them while they are eating?
That would explain a head covering, but it’s not sufficient to explain a fedora. Traditionally, the guys who wore the fedora outside, also wore a yarmulke under it – so that when they came inside and took the fedora off, they still had a covered head. An orthodox jew who (a) holds to the belief that the head should be covered at all times (not just at prayer or outside) and (b) wears a fedora, and (c) doesn’t wear a yarmulke — still looks like a hipster to me*
*I admit that I thought the rastafarian look – knitted scull cap, dreadlock peyot, was a pretty cool way for a teenager to be both orthodox and to offend his parents 
If I’m wearing a broad-brim hat like a fedora, I definitely take it off at the table.The brim cuts into your eyeline, and sticks out behind your chair. Wearing a fedora inside is like not putting down your umbrella --sure, there’s no law that say you have to put down your umbrella when you come inside, but after you’re 25 you grow out of that kind of nonsense.
Etiquette rules exist so we don’t have to reinvent what to do in every situation. They are just societal norms. Some things do become outmoded (I no longer store my buggy whip outside in the barn). They’re not laws, just norms. Do you say “please” when you ask for something? Why isn’t that archaic? For men, uncovering the head (when not otherwise required by religious practice) is a sign of respect, but not required outdoors because presumably you need it to keep your head warm or shade the sun (a tip of the hat will do). If you want to keep your hat on, feel free. Nobody will stop you. Except this guy.
Respect to… who?
In general I agree with your overall explanation and sentiment. Ref this nice tight tidbit:
Agree that’s what they’re for. But there’s a second factor.
In a society (or subset thereof) that expects and demands a high degree of uniformity, there does need to be a rule for everything that everyone knows so everyone’s behavior can be uniform as demanded.
In a society that is a lot more relaxed about behavioral variation, the need to “know what to do in every situation” is much less. Just do whatever and it’ll be accepted.
Said another way, the desire for etiquette is the flip side of the desire for conformity. And vice versa for the lack of desire.
Clearly our society is becoming more diverse. And much of etiquette is just roadkill on the way to acceptance of greater diversity in behavior.
I wouldn’t wear a hat in a restaurant with tableside servers but I don’t wear hats except for utility, anyway (cold wx, helmets, low sun while mowing, sun protection, etc). But I’d certainly take notice of a man wearing a fedora at somewhere like Everest but nothing beyond that. What, am I supposed to take him aside like his fly was down?
Why is that a sign of respect? It is arbitrary. Why not uncovering ankles?
Yes, in those days, a gentleman would not dine with his at on. But a lady would not leave home without a hat, and would leave it on to dine. I would say, if the restaurant has a hat-check area, take it off.
In those days, a man always takes his hat off in a public building, like a library or city hall.
Is this a good time to point out that at the Passover seders I’ve been to all the men were wearing head coverings of some sort…?
I’ve known a guy named Clem for about thirty years. For the first twenty five of those years I never saw him without a baseball cap on his head and I thought Clem was his real name.
About five years ago I learned his name is Steve, and I saw him take his hat off for the first time. I was shocked to see he was bald! He has a complete head of hair except for where a baseball cap covers.
I am just describing the convention. I didn’t invent it, and it goes back centuries. I don’t know, but an etiquette scholar might.
Yes, I think it is. Although I implied that above.
At a formal religious occasion, like a Passover supper, even the less observant men are likely to be wearing a head covering.
Once, my husband and I attended the dedication of a bridge, and a brief prayer was offered. Every other man there (with a hat) removed his hat, and my hatless husband put on his bicycle helmet, because he was taught to cover his head when he prays.
Which wouldn’t be a problem if your napkin was properly over your lap.