Why does borrowability of shares inversely correlate with probability of assignment or exercise?
I disagree with the author's premise. The primary reason that options are exercised early is because of discount arbitrage (an ITM option trades below its intrinsic value) or there is a dividend arbitrage in an ITM put (the time premium is less than the dividend, NOT the call).
No one with a lick of sense would exercise an option that has time premium remaining because that would be throwing away the time premium and incurring more transactional costs (B/A spread and commissions if you're still paying them).
I listened to another minute of the video and I think that the author really has no clue. He talks about the possibility of an OTM short option being assigned early because of a dividend. Now who wouldn't want that to happen to them? If the stock is \$102 and I'm short the \$105 call, please, make me sell the stock to you for \$105! It makes no sense at all.
He continues his errors with another example. A stock is \$101 and you are short a \$100 call that's worth \$2. The company announces a \$2 dividend and the call owner exercises and you are assigned. Why in the world would he do that? He could sell the call for \$2 and buy the stock for \$101. Exercising throws a dollar of time premium away.