[a-priori]

The best online merchants these days are about successfully mixing online and offline experiences. The term of art here, if you want to Google it, is 'omnichannel': the idea that their experience with the merchant should seamlessly span all the channels (online storefront, brick and mortar storefront, support channels, etc.). One major use case for Stripe Reader would be as a payment solution for omnichannel merchants.

Omnichannel commerce is a product of the Internet, and specifically an Internet that's capable of supporting commerce, because a business could not operate this way without it. Seen through that lens, Stripe is "increasing the GDP of the Internet" here.

Though, I admit, this is starting to stretch the limits of how much you can consider "the Internet" being a separate place.