Absolutely an open hot start plunger will cause cold starting issues. A cold motor wants a rich mixture which the choke accomplishes. A hot motor wants a lean mixture which is what the hot start accomplishes. So, with the plunger open it was leaning out the mixture...just the opposite of what a cold motor needs. So, good find and fix. If you don't already have one get our billet aluminum hot start nut. The stock nut is plastic and prone to break causing these types of issues.
Yes, you can always run it until the valves move making it difficult to start. The only problem with this philosophy is it will determine for you when you can't ride your bike instead of you determining when. If you have a race or epic ride coming up it's best to pick the best time to resolve valve issues.
Ken