English is my 3 language so i could be wrong, but i thought Skin in the game means exactly that, there is skin to be lost if shit goes bad.

in the example of "Do you know why surgeons don’t perform surgery on close friends , relatives , or spouses?" the skin in the game would be that you f up and do something bad to your friend... Or if you do it as a job on person that you don't know and you f up skin in the game is that you will lose your job and not work again as a surgeon.

not having skin in the game would be if a surgeon is doing an brain surgery and you walk in as a random guy (not working there, no one knows you, you just ended up there and have nothing to lose) and you are convincing the surgeon that he is doing things wrong and he does not know how to do his job.

Does that mean the the surgeon is 100% right or that you are 100% wrong? No. but does that mean that surgeon will get fired if he listens to you and fucks up and you will just walk away with no consequences? 100%