Very bad idea. I understand wanting to ask for advice from a specific person, while maintaining your anonymity, but honestly I think way more people would use it to harass each other. That would be a very easily abused feature.
Personally, I don't think it'd make any difference because it's very easy to make a new account and earn 150 xper to reach lv2. And you can earn 100 xper by just connecting your Facebook and Google accounts.
With the new account, you're absolutely anonymous for individuals because who knows who the person is behind that xyz username.
This anonymous messaging will increase the harassment but you can always allow from the people you follow and then you can have an idea that the guy who sent you message is one of the people YOU are following.
You're right, it has pros and cons. But cons side is heavier and people aren't remotely interested in this feature.