Harry's usually way better about exhausting every avenue at his disposal. Here he didn't really even try.
Maybe that was the point. That Harry was pretty thoroughly defeated by Murphy's death?
Exactly, Harry is usually better about exhausting all avenues. That's why it was so weird to me that this guy that tried and tried and tried to find a solution for Susan's problem, wasn't going to even ask Vadderung —who he apparently can just call up on the phone— about Murphy's wellbeing, I mean, he could have just ask, I'm not even talking about trying something else, like ask for the details of the process or hell, even try to negotiate something. And it's not that he should have, just that's more according to his character.
The same thing with the Thomas situation. I liked the drama of it all. But you are right, it was pretty much as if Harry didn't even try anything. And then, he was also just fine handing over his Nephew, his own blood to these other people to be raised —and Lara is just fine giving away her grandbaby as well— For always having a speech about "family" that was weird. No wonder Thomas was upset with Harry, and personally, he should have stayed mad. Though, tbh, I'm not buying the act yet, I think we should expect something else from Thomas, he's not over his anger toward Harry.
I think we're supposed to believe that Harry not trying to find a loophole for Murphy's Einherjaness is meant to be a sign of maturity or something, the same with the Thomas situation. Either that, or Harry was really really completly broken. Maybe it would have been read more like that if we'd seen Harry try to do something regarding Murphy's death and failing miserably (of course) and that would seep into everything else.
There's also a point to be made about how much of Harry's mental/emotional state actually was about Murphy and how much it was about his shame over his actions right after. I felt we spent more time with that actually, than with Harry lamenting the future he lost with Murph. But well, that's another matter.
But my point is that these things are important maybe in what Harry didn't do with the situation with Etri, I remember thinking that the matter just disappeared from the page rather quickly. And then they just hand the baby away. Yes, they know that he's going to be treated as a prince, but still. On the other hand, Harry is mostly comfortable with offshoring parenting Maggie for a long while, so maybe is not a big deal for him. But isn't this kind of thing what Dresden fights Eb about, that he will make the choices for his child? And he ultimately brings her home with him, that's the "resolution" for the Maggie situation in this book. Harry just took that choice from Thomas, in a way.