I read their blog but the points they highlighted don't mention when BitPay takes that BitCoin and exchanges it for Fiat. Let's use PayPal as the perfect example. PayPal has MSB's licenses because they can convert your local currency into another local currency therefore besides acting as a payment processor they are also facilitating an exchange.
Like mentioned above an exchange is taking place, unless it can be argued that that exchange is occurring on a regulated MSB licensed exchange and their regulatory power is being used, which would take liability away from the "User"whether it be BitPay or the Merchant acting individually as a Money Transmitter.
Here is a list of all of PayPal's State Licenses: https://www.paypal-media.com/licenses
You can see most of them are MSB Licenses for the States requiring them, I doubt PayPal got them for no reason if they weren't needed.
Also it might be that PayPal got them to umbrella it's users under their MSB licenses, if the responsibility falls on the Users since they are facilitating the activity through PayPal, but this would not make logically as that means PayPal is classified as an MSB to have any of those benefits apply to it's users.