no one of a sane mind is carrying emergency lithium or any battery as a matter of fact . to check if your battery is charging, you need to use 'voltomiter'. when you pull the throttle, the V should be above 14 or else it is not charging properly.On one of my bikes, a Husaberg 550, I noticed that starting improved significantly when I fitted a new battery. The engine would just 'catch' almost immediately compared to before, when it would take much longer cranking, even when the old battery was still ok-ish.
So the voltage of the battery influences the ease of starting (stronger spark). This could explain on bikes with weak, sub-standard performing stators why cold starting can be problematic: weaker battery voltage, coupled with less 'help' from the stator's current when starting up from cold.
I suppose the way to test is by checking the output of the stator/measuring the voltage on the battery with engine running. If you have difficult cold starting, maybe have the stator checked at the dealer before the bike leaves you stranded. And, all of us, carry an emergency lithium battery?