Some patients crossed between the two groups, but this does not matter, as they were testing the effects of the initial assignment.

It matters to your case. I refuse to believe that writing a patient's name on this list rather than that list has a direct causal influence upon their state in 2 years. The influence can only proceed via their actual treatment.

Assignment ---> Actual treatment ---> Outcome

The decision facing you is whether to have surgery early or not. That is the thing whose effect on the outcome you want to know. To the extent that in the study this differs from the initial assignment, the study is diminished; therefore it should matter to the people conducting the study also.

I see from the paper that 23% of those assigned to Watchful Waiting nevertheless had surgery within 2 years, and 17% of those assigned to surgery did not have surgery in 2 years. (Some others died of unrelated causes or left the study early.)

I'll leave it to a dan-grade statistician to judge how to obtain the best conclusion from these data.