Why are you trying to spread that falsehood? How is dogecoin "already the cryptocurrency of Reddit"? It wasn't created by Reddit, no one in Reddit is using it (except r/dogecoin and related subs), and it's supposed to be a joke. I don't think Reddit wants its shares to be a joke too.
They can create their own crypto-currency on top of an existing one, using dogeparty or counterparty. This is most likely their best bet, because they won't fall into a lot of the security/hashrate traps with a lot of new coins.
The fun thing is that as soon as I saw the plans to distribute ownership earlier today, I figured why not use their own cryptocurrency? I guess I'm starting to predict the future, albeit only a few hours in advance