What, if anything, is the role of CSR and/or brand purpose in QSR?

In a nutshell

Brand purpose and CSR can play extremely beneficial roles in QSR, when the appropriate time is taken to strategize for them.

Brand Purpose

There's never a downside to creating a brand strategy. Successful branding can provide so many benefits to a company, it can provide the brand more recognition, enhance brand loyalty, and create greater differentiation in the brand's category. Branding allows QSR establishments like Chick-fil-A to be more than just another fast-food establishment, it's THE place to go for chicken but not beef.

However, it does take time and effort to create an effective brand strategy. Without the backbone of research, brand strategy can't possibly be effective. If brand idea isn't true, relevant, resonant, and differentiating, it can come off as insincere. This can also happen to poorly incorporated CSR plans.

Corporate Social Responsibility

CSR can be a great move to improve the overall image of a brand, and create more monetary gain for them. TOMS skyrocketed their success with their "one-for-one" campaign (which later was found to have complications). How much this actually effects the bottom line in these companies is up for debate. However, in my experience, CSR can seem disingenuine when it's executed poorly. Like Shell claiming they care about the biodiversity in our Oceans?!

Conclusion

For companies in the QSR space, they should always put in the time to create a successful brand strategy, and if CSR effectively fits into their Brand's purpose then go for that too. Just don't do more than what your brand should be tackling, your consumers will see right through it.