You are conflating three things: skill, formal authority and personal incentives. A doctor would operate on herself if possible, if the alternative were a layperson or a much more incompetent doctor for example. A judge who is corrupt may still be better than an idiot who doesn't understand the law at all in many cases.

Skill levels and formal authority being equal, things can go either way.

Remember, a stake can also make you more inclined to be careful and diligent, not just cloud your judgment. Disinterested and uninterested are not the same thing. "Disinterested" is when you have power based on position (ascriptive authority) but adopt a professionally neutral position (which takes skill, like John Roberts in the Obamacare case) without being sloppy (a potential consequence). A student who has much to learn might devote more care to a routine case than a senior doctor who finds nothing interesting about it and gets careless.

So you can conclude that in situations where skill levels are equal, and the case does not require passionate interest for successful application of the skill, and there is some benefit to dispassionate disinterest, a third party is better.

This is usually only the cases (and almost always the case, so almost necessary and sufficient) when formal ascriptive authority is involved. Hence your instinctive use of doctors and judges as examples, rather than cops or teachers.

These situations are actually quite narrow, formal and specialized, so in the general everyday case, I'd expect stakeholders to make better decisions because they are likely to know more relevant things than random third parties.