I think we've wandered away from the OP's intents but...
along with stiffer frames often the wheels are smaller (17") & steering centre to fork offset less, reducing trail. also steering castor / rake may be steeper. All things that make a bike fall into a corner (or not). Re. offset; Olde Brit Bike's fork to steering stem layout is quite triangular looked at from the top yoke (not yolk, runny forks would be a joke). Modern bikes are very flat in comparison, the forks & steering stem are almost in a straight line.
I imagine the current crop of GP teams are trying to answer Hillbilly's last question! Those bikes are so stiff & precisely set up & still they don't always work how the rider wants or likes. Too stiff, too much flex, removable frame sections in order to change stiffness or flex etc.
Olde Brit bikes; You can spend many hours acheiving perfect alignment of parts & some of that effort will pay off. The garden shed mechanic can assist this by fitting say, taper roller steering bearings - things that make the handling feel more exact / controlled but often the bike will have some corner cut design-wise for all sorts of manufacturer's reasons, not the least of which was financial. My old road going Goldie had the silentbloc bushes in the swing arm when I first got it. It handled horribly in corners until I replaced them with new ones which felt only slightly better. My race bike had actual bearings in there & handled much better but then again it was a lot lighter, none of that heavy chrome stuff on it for a start. These days our tyres have way more grip than in '59 & can grip better than the bike can cope with. Certainly they ask questions of the bike's suspension & handling.
Also some frames as a rule just handle better than others. Does anyone favour a Triumph twin over a Norton
featherbed frame? Anyone who's ridden a bike with a bent or cracked frame will notice how it tips into a corner easier one way than the other or sets up a weave in certain conditions. I've ridden a few like that & also a few that were very nice to ride - ones which had a lot of care taken in setting them up although I'm pretty certain those responsible had access to nothing more technical or accurate than a vernier caliper & some straight edges.
In response to previous comments, it's usually worn or misaligned chains that break although sometimes extreme circumstances cause breakages - Ballaugh Bridge IOM being a case in point. That is a rear chain breaker, not what this thread is supposedly about!
Getting good belt alignment has got to be worthwhile, I was probably just lucky with my first b-drive setup on that front. Whilst I agree that the surface table is a wonderful thing, probably 99.999% of old bike fettlers don't have one or access to one or the money to pay someone who has one, to set up their bike / frame. Maybe we might once in a lifetime actually splash out good money for an expert to fettle a bike or frame for us but I bet it's not common. Some of the big money classic racers may go down that path more often.
We can at least use the other accessible methods that have been suggested. That said, I bet many of us have just bolted a bike together, trued the wheels with a straight edge or some string & then happily thrashed the bike around for a few years, adjusting our riding to suit whatever quirks the bike has. Probably an attitude that contributed to the Brit Bike's reputation for sloppy engineering... ;-) Most "Classic" road riders are not pushing the limits of handling on their Sunday ride.
Vice like grip is not how I ride. There was an issue with Ducati Hypermotards (& probably other quick, sit up & beg bikes with wide bars) with riders complaining of weaving at speed. Almost certainly due to rider accelerating hard & hanging on like grim death.. The wide bars exaggerate any rider inputs - as soon as the rider imperceptably pulls the bar one way when "hanging on" they "correct" the movement the other way but overdo it, leading to repeating the mistake the opposite way & before they know it there is a lovely if slighly lairy 100mph weave going on... Holding on lightly tends to prevent it & is a more relaxing way to ride.