For example, in the sentence:
He eschewed his father's profession.
does this have the implication that he found it somewhat repugnant, or does it just mean he decided it wasn't for him?
I think the word eschew has a moral connotation.
eschew means to avoid doing something, especially for moral reasons; deliberately avoid using; abstain from. Synonyms include to refrain from, give up, forgo, renounce, etc. You eschew things that you find morally or aesthetically wrong, or that you have chosen to find wrong.
Good people have eschewed violence, such as Gandhi, Martin Luther King Jr. and Christ. Mother Theresa eschewed material comforts. Addicts can decide to eschew drugs.
To stay in good standing, members of the Society had to eschew gambling.
Googling eschew + (a vice) will get you enough hits. But googling eschew + (a virtue) will get you fewer pages.
Machiavelli encouraged his prince to eschew virtue if doing so would keep the prince in power and maintain order.
It is likely that the son saw his father's profession as something to be avoided.
It means to actively avoid something. In the context you've presented it, it implies that he avoided his father's profession due to a disdain for it.
You wouldn't use eschew to mean he merely wanted to be a fireman, not a police officer like his father. You could, but it seems harsh.
It would be more that he avoided his father's profession due to the pitfalls he associated with it.