We have touched on several times in this thread, it is more than likely not getting enough fuel or not being enriched enough by the choke when it's cold.
Having said that, I will say that the horizontally opposed twins can often be slow crankers but the coil system on these engines is pretty strong and even spitting one by hand will spark a spark plug so if it's turning over and spending the engine it probably has enough spark to run so I'm still guessing it's a fuel or enriching problem.
The answer to the fix is to find out why it's not getting enough fuel up into the throat of the carburetor when it cranks over.
This is why you start by checking the choke butterfly.
It's not common on these particular engines but these engines are also extremely old at this point and it would be rare for one this old to have not had throttle cable removed, replaced, carburetor off, etc or cable and levers and stuff just move around a little bit over the years so it's not pushing the choke butterfly closed far enough.
There was a whole rash of Kohler single courage engines a few years back coming from the factory with the chokes not adjusted to close far enough. Fix was as easy as using a torx driver on a purple paint marked screw to increase the choke movement but lots of people found out they couldn't get them to start when they were cold unless they remove the air filter and sprayed a little gas or carb cleaner or something into the carb and then they started right up and ran perfectly.
So it certainly happens.
If the choke is closing all the way, then you start considering the level and the float bowl is too low due to leaking out or seeping or whatever.
If this happens after only several hours or just a day or two then it would have to be a leak. If it happens over a week plus the gas level could be evaporating in the bowl but that's not going to happen just overnight.
Someone mentioned the carb solenoid not having enough voltage during cranking to stay open but I can't see I've ever seen that and would be highly unlikely.
Those things don't take much current at all to stop open so if the electrical system is working properly that's probably not the issue.
HOWEVER, I have seen a couple of situations where the ignition switch was messed up or wired improperly and the carb solenoid had 12 volts when you click it the first notch to the on position but it did not have power at all when you were cranking.
It would be worth checking this on yours and would be a very quick easy test. You could just use a test light or voltmeter on The wire pigtail that goes to the solenoid and watch it come on when you click your key to the first on position and then make sure it stays on when you crank the engine and hold the switch in the crank position.
Strange things happen with old ignition switches especially when they've gotten wet or when the key has been pushed and pulled and jostled around by hitting things.
I've seen a flaky ignition switch kill multiple ignition coils because it somehow internally allowed voltage for just a split second to go to the terminal which connected the kill wire from the coil to the chassis when in the off position. Voltage to a coil there will kill it quickly if not instantly.