This has worked so well (I did say that Starbucks' app is the most popular in the country) that I wouldn't be surprised to see a similar gradual process at work as Starbucks tries to get customers to use the app in other stores. What seems likely to help is to be explicit about the benefits of using the Starbucks app, both for consumers and merchants. That might include discounts, at least at first, and that means some serious multiparty negotiations will have to be conducted. Starbucks could sweeten the pot for merchants by offering some cross-marketing opportunities: Signs in those nearly 20,000 Starbucks stores might say something like, "Use your Starbucks payment app here today and anytime with these fine local establishments." Something similar could be generated on Starbucks' website based on a user's IP address, geolocation data or a ZIP code entered six months ago.
If anyone can figure out a way to make all of this work, it's likely Starbucks. After all, this is the company that routinely gets people (myself included) to pay $7 for a cup of coffee.