"While somewhat taken back by his stubbornness in repeatedly insisting he'd done nothing wrong, he ultimately came to respect his son's point of view".

Shocked, astounded, disgusted, fed up with, sick of etc. would be too strong here I think, especially given the moderating 半ば term ("half" or "somewhat").

I'll put this out there as well.

In the context of a father and a son, and the idea that the son has repeatedly denied blame, 半ば呆れる could also mean to "give up on" in regards to a person.

First Person: Toward my son's resolute assertion that he had done nothing wrong, I had mostly given up and, in the end, accepted it as a quirk of his personality.

Third Person: Toward his son's resolute "I have done nothing wrong," he had mostly given up and, in the end, accepted the continued denial as a quirk of personality.

Mine is a bit of a literary interpretation, but having more context always helps. Is this a book? A movie? Something else?

I would also love to hear a native Japanese speaker's interpretation of this.