Setting your hat crown down is really more of a western hat thing, so as not to distort the brim shaping. It's fine, even preferable, to put a soft felt hat like a fedora brim down to prevent distorting the crown. Realistically, you're probably ok to put a modern fedora crown down, there's more stiffener in the felt these days. Though I stopped putting my Stetson Fortune and Glory crown down after I thought I saw the beginnings of a flat spot (and had coincidentally seen Matt Deckard posting about it).
I think folks were generally a bit less precious about their hats than we are today. Nowadays, one might save up for a really nice beaver hat because they're widely considered the best quality available. But I think this gives us modern folks kind of a distorted sense of price. Not only was there far more variance in pricing, one didn't necessarily need to shell out for the top of the line hats, because good quality hats were available at a lower price point. Contrast the $400-$800+ that we might spend vs a 1950 $5 Stetson Playboy (a best selling hat) that would be about $60 in 2022 dollars. A $15 hat (higher quality 3x beaver or 'Fifteen' grade) comes out to $177 in 2022 dollars. It's not exactly a direct quality comparison to today, but there have been a handful of Australian Stetson Open Roads (made by Akubra)
posted to the Fedora Lounge, and those were 3x hats. Of course when the license ran out, they just started calling it the Campdraft. But as a price/quality comparison, it's interesting that Akubra is still more or less situated the same today.
An all beaver hat at that time really would have been gilding the lily. The Stetson 100 (which came in its own luggage-like locking case) works out to a bit under $1200 in 2022 dollars. Not exactly an everyday hat! If the mid grade Stetsons were as good as they used to be, I wouldn't care so much about beaver content.
So when we consider that the hats most folks wore were relatively less expensive; that they were just a normal item of clothing; and, they could easily be refurbished, I can see why folks might have been a little more nonchalant about grabbing by the crown. And that's really more like a 'best practices' thing moreso than a hard and fast rule. It's not as if you'd be socially ostracized like wearing white after labor day, or
get beaten in the street for wearing a straw hat past the 'appropriate' day.