"On the other hand, all the leagues have significantly lower average hotness in the first half compared to the second half, so maybe it’s not just the NBA that has a boring first half problem."

This seems to be a misconception. It stands to reason that the chart will grow jumpier as you near the end of the game. In options lingo, the implied volatility will be more stable the further you are from expiry (the end of the game). Theta decay and all that. The market is basically giving you an integrated forecast from each point in time until the final buzzer, and as that window shrinks you expect the odds to be jumping around more.