The My Site thing is a social profile thingy. From what I understand it's enterprise social media that lets your staff do lot's of things including the ability to share docs with people and it may even work with the task functionality.
I agree with your managers- this is an important tool for staff to use. It's just not as black and white as do they need it?
I think it's very interesting that this organisation actually wants people to use it. From what I've seen many organisations freak out at the idea of people having their own space and having the ability to post things that they can't control.
SP2013 out of the box isn't that bad- 2010 was revolting. 2013 has a lot of new stuff that significntly improves the user experience. Search actually works! LOL
From a workplace culture and collaboration perspective, I think it's nice to see that management want this. I personally think it addresses multiple needs that all organisations have; the need to collaborate with your staff and co-workers, networking, exploring new ways of working. It also works as a people directory.
I think the My Site should be allowed. People don't actually have to do anything with it but the option should be there. We should have multiple channels to connect with the other humans in our workplace and that could never be a bad thing. If they don't all use it- so what? Is something bad going to happen? Probably not.