Shameless self promotion: this is the exact problem my co-founder and I are working to address: https://www.perspexilabs.com/ (site currently looks broadly-focused, but we're targeting medical).
We're finding patients are actually very happy to give feedback, especially when given a neutral medium to discuss their experiences and when doing so very quickly after an interaction. The super-tight feedback loop not only makes everyone more satisfied, but it also has the chance to improve quality of care too.