You're being awfully judgmental on very little facts. You don't know what efforts he made to find the mother, what efforts she made to stay hidden, etc. You seem to take it for granted that he could have found her fairly readily and that may not have been the case. If he expended considerable effort and came up empty then there comes a time when it's reasonable to give up the active effort. Let's stick to the legal situation and not judge a situation on incomplete facts.
"[H]e gave up." Judgmental.
"[Y]our valiant husband decided to just give up and move on." Judgmental.
"I don't believe for one second that your husband didn't have the ability to figure out his various options . . . (and neither should you)." Judgmental.
And every last bit of this is completely unhelpful to the questions asked.
Dad should request visitation based on the distance between him and the child and use every bit granted. It is mom's responsibility to make the child available; he should immediately file for contempt if visitation is denied. Step-mom should be welcoming and do all possible to make the child comfortable in the child's additional home.
As I stated before, you made judgments based on relatively little information. There is a lot we don't know here.
But some of those resources are not cheap to use, and he may not have had the resources to pay for it. And the ones that are available aren't always very good. Even with good resources it is not always possible to find someone who does not want to be found. I know this from having tried to track down people myself. I don't know how much experience you have with doing that, but you may be assuming it's easy to find anyone. It's not always so easy to do.
Your statements come across as in fact judging his motivations. You may not think so, but the fact that I and others do suggests you aren't seeing your comments the way that others do.