The Rav Kook, in his book Ein Aya on this segment (9, 221), reads it differently from your translation (in my opinion, this also fits the verses that were brought as "proofs" better).
The term "יצר הטוב שופטן" is explained by the Rav Kook as "the yezter hatov serves as their judge" - not that it judges them, but that their judgement (of other things) is managed by their yetzer tov.
We judge everything with our mind - so our mind is the judge. However, the Rav Kook explains that this isn't exactly true. We do not judge things coldly and purely logically. All "ענייני המוסר", issues of morality, cannot be clearly and coldly cut and judged by the mind. Eventually, it comes down to one's "יושר הלב" - the "moral health" of one's nature.
So a person's moral feelings are intertwined with each and every moral judgement that his mind decrees. If he is righteous, his "judge" will be his yezer hatov - his good morality. And if he is evil, those evil feelings will affect his judgement, and his "judge" will be yetzer hara.
One last question. Since most people are "בינונים" - between righteous and evil - they have both yetzarim affect their judgement. So maybe in this case we can say that their "judge" is the pure mind - it has both sides and therefore can decide logically.
But no, this is not true. Morality cannot be judged in itself. It is always connected to the inner feelings and nature of the person judging. So they too cannot judge things 100% with the logical, pure mind. Both yezters will affect their judgement - both their good and bad inclinations. Therefore, obviously, it is better to be righteous, so that one's judgements can be as true as possible in our life.