From what I'm seeing you seem to already have the clusters. If there are arranged in points then follow the instructions here:
This will give you the number of points inside each buffer
Then simply open the field where your sums are located and create a new Float field. Then using the field calculator enter the percentage (sum/number of total surgeries)*100
And that's all there is to it.
Forgive me if I'm not understanding but just from a data stand point, I think there might be an issue if there are overlapping buffers and points are being counted twice. Not so much that it will throw the percents off, but the fact is the theoretical patient didn't go to two hospitals, they only went to 1...So individually your percent might be correct but comparatively they will be off.
Clip overlapping polygons. There's plenty of ways to do this that are just a google away.