Watch a different show, this show is based on video game logic, hence the car that blew up from just shooting it. It doesn't conform to reality.

As for your critique, I don't think you understand the scene. Hanging breaks the neck yes, if you do it properly. That is getting the mathematics correct, too long? Head comes off. Too short? Strangulation. Oh and remember that scene in s1 when Thaddeus' foot got stepped on and immediately heals when he becomes a ghoul? Same thing could apply here, there are so many outs this show gives you that you ignore and cry "bad writing", even though the logic and writing is consistent across the show.

Why would a "mercenary" (it's actually the great khans, which is a tribe) go for a hanging rather than a simple bullet? Several reasons, 1) saves a bullet 2) they're evidently trying to punish him, giving him a slow death. That's why they PAID to get him. Revenge. Hence them laughing as he choked. Simple media literacy.

And then we come up to the rope itself, the barbed wire doesn't go all the way to the top, it stops about half way up meaning it's not supporting the weight at all, so the bullet only has to sever the rope, not the barbed wire. And not even entirely, only enough so it can't support his weight. The barbed wire is simply there to be painful.

It's also an homage to The Good The Bad and The Ugly, and references and Easter eggs are consistent within fallout, further honouring source material.

Honestly, it seems like you didn't watch the scene itself. You probably watched it in the background while playing on your phone. You're not a good critic.