other leagues aren’t as fierce about taking things down
Because it works. It drives traffic to MLB.com and raises their advertizing revenue. They’ve decided that works as a business model.
What part of “expressed written permission of Major League Baseball” is confusing? 
yeah well if that’s the case why doesn’t the NFL and NBA and NHL demand youtube take down their clips too
They’ve decided on a different business model. But I’ve seen them take down clips on Youtube as well.
So if all I have is implied oral permission, I better head for international waters, huh?
It’s possible the other leagues instead run ads on the YouTube clips.