Adding a new tank class wouldn't change anything on availability. The population of people who don't want to tank, simply don't level a tanking class. The same goes for a healer class.

As others have pointed out it's nothing to do with FFXIV, pretty much every MMO that has traditional roles run into this issue.

A simple answer is if you don't want to wait around as a DPS for a healer or tank is simply level one yourself.