You won't have to worry about the bike doing a somersault with the poor brakes.
It's actually pretty hard to cause it, even if the front is loaded up compared to the rear. I've only managed to nearly do an endo on a touring bike with the front loaded up while going down and hitting a speed bump on a hill and while under heavy braking.
The leading cause for a somersault on a bike is because the bike stops, but the rider wasn't holding onto the bike wel. This causes the rider to crash into the stopped bike, whereby the fulcrum of the front wheel will cause the forward motion to turn into an up and over motion.
On the other hand, some poor road conditions, like gravel on pavement, do cause the front end to wash out under the rider, even while going at a modest speed. Wet cobble stone and brick pavements are also not pleasant to ride on. But on some asphalt or concrete pavement, such speeds, even when loaded up, will not cause a wipe out. It's only when you ride over a painted line or manhole cover does the traction suddenly vanish the rider ends up on the ground, even while going in a straight line.
Not only are the differences in road conditions different from 1929 to modern day, but the tire and brake technology are also much improved from the past to present. With all these variables, you can end up with a large variety of what is rideable and what is not. If there was an old version and modern version of the postal bike, top speed could be a spot to give them a difference.
I do ride all year around, so I can give a pretty good breakdown of what is possible with a bike. I sure as heck don't want to try going upwards of 25km/h riding a 1929 bike on cobblestone or muddy cobblestones. But give me a bike from at least 1980 and I'd do it.
but anyways, the postal bike is one of those bonus vehicles in the game, where one would only use for the sake of completeness. These bikes get overwhelmed quite fast, because they can't keep up with the capacity demands. And even some of the cars and vans have trouble with this as well.
I think the important part is trying to strike a balance between realism and play-ability of the goods road vehicles, because they're basically unplayable right now.