That's actually illegal.

People, the EAD is a federal document and please don't be foolish enough to bring with you everywhere in order to use it as an ID. That's not the main purpose of this document, and just like your SS card, you will have to file several forms to get it replaced if lost or stolen. EAD is also several times more serious of a document than a SS card, as it is a valid form of proving your identity, age, employment eligibility and signature. It is one of the very few List A documents available, and is considered on the same level as the Green Card and the US passport in terms of proving the four aforementioned factors.

Also, there is no reason why getting a learners permit would take longer than getting a state ID, so the latter should only be gotten if you have no use for a DL.