Why does "skin in the game" matter? Why is that a useful signal about whether this company will be a good employer or not?

It's not a useful signal about if they are a good employer, but it is a good signal for how likely they are to hire you: asking everyone who applies to take several hours out of their life to do your test is free. Having someone from your team interview them for several hours very much is not so you'll be more careful selecting candidates you might actually hire instead of just taking a shotgun approach.

For me it's a useful signal, it means they value my time equally as theirs. I'm happy to do the take-home test after the onsite one when they only testing that I really know what I said I know but I'm at the finish line. A lot of companies require it right after the phone screening(usually with an HR person who knows nothing). I burned myself that way before. After finishing the take home test on the on site interview after 5 minutes it was clear they were looking for an ops person advertised as DevOps.

Right, having the employer invest something means they'll be more likely to check for a good fit before reaching for homework as a filter.