You can't just go by favorite or not. The main factor is how big of a favorite. They should take the money line Vegas puts out and analyze that.
Example: Vegas has GS as a 3-1 favorite to win the series (-300 money line). How often does 3-1 favorites actually win in each of the sports?
Where in that whole video did they mention favorites once? It has nothing to do with favorites. Statistically speaking, if there was absolutely no skill involved, you'd expect a certain distribution of the teams W-L records based on how many games that sport has. Fewer games means a wider spread of distribution from .500 records, and more games means a tighter distribution of W-L records. If a sport has a wider distribution of W-L records than you'd expect based on the number of games they play, that suggests that skill plays a larger role in that sport than in a sport where the distribution is pretty close to what you'd expect from its number of games.
ETA: I realize that the title mentions underdog, but that doesn't play a part in the calculation. Once you've determined that luck plays a larger role in some sports, that intrinsically means that more underdogs will win - hence the title.