Actually, Valve
doesn't own the DotA trademark. More information on it can be found
here. And if you want proof, look
here. Go to the Tademark Electronic Search System, use Basic Word Mark Search and type 'dota'. Note, there are multiple registrants and Valve's doesn't have a registration number. Meaning, Valve can't send out cease and desist letters to anyone. At least, not yet.
What's interesting is that Blizzard themselves aren't filing for trademark; especially since they're the ones who technically own the map (if I'm not mistaken, since that would be what allows Blizzard to take down user-created maps containing inappropriate content).
Thing is, that was not an option. The people who requested Youtube to take it down was not the owner of Blizzard but an employee of the legal team. Therefore, they are not in a position to use their own discretion and must instead follow company guidelines and protocols if s/he values his/her job. As to why go straight to Youtube? Because Youtube was the one hosting the infringing video; and thus the one who is ultimately responsible.