Tons of small businesses in San Francisco use Pay With Square already, and they don't have the rigorous employee training procedures that Starbucks has.
I use Pay With Square all the time—at tiny bakeries, big coffee shops, and restaurants. No one will look at you lost.
It's convenient because I don't need my wallet, I don't need to sign a receipt, and I don't need to do anything other than order and walk away.
It's a humanizing experience actually. Try it once or twice. My mind was blown the first time I paid with my name using Square, and since then I have explicitly sought out coffee shops which have it, because of the quality of the experience.
Why would they look at you lost? If Starbucks is implementing this they will have to train their staff in how to use it and paying by saying your name is one of the best features of the product. It will be great for Starbucks as I assume it is a much quicker way to checkout than processing a credit card and, at least where I'm from, Starbucks always has long lines.
I don't see the value for Starbucks customers either.