Wow, I completely disagree! I loved this moment and thought it made sense and fit well. As you said, this is someone he's spent a long time with. He is extremely close to Aux, and at this point they're even more linked than a normal Nahel bond, with Aux being half-dead and only able to sense things through Nomad.
Nomad's used to living life with Aux there, so when he's in a life-or-death situation, he of course turns to his best friend and magical partner. That is his instinctive reaction. I can easily imagine myself doing the same thing; if one of my best friends died, someone who I share thoughts with and am often talking to, I can see myself excitedly clicking into a chat...only to suddenly remember that they're not there anymore. I can't talk to them about it.
The human brain can only hold so many things in front at once. (And I do believe that psychologically, Nomad is still essentially human, if warped by the Dawnshard.) He was in the middle of an attack on the Cinder King. He was thinking about the upcoming fight, Elegy and Rebeke, his Shardplate and how it was gone again, the overarching plan...he's had a lot of different things to focus on throughout this book and not a lot of time to process more than a few. Aux died, what, an hour ago max? This isn't something he's internalized yet.
He has a plan to execute and another day to survive. Some thoughts and emotional processing aren't at the forefront yet. (Which I think fits pretty well with the overall theme of the book - he never stops running, moving, fighting. The pace is fast and he doesn't have the time to process, as much as he may want to, or he will die.)
On top of that - people just deal with grief differently! Some people, when someone they love has died, completely fall apart. Some people dissociate and avoid like hell. Some people face it head on and deal with it easily in the moment, but later on it hits them much harder. And some people hurt, but their grief doesn't damage them, even if they felt deeply for the person who they lost.
I have lost people; thankfully, no dead best friends or immediate family, but I've lost people I cared about, and...while I missed them, I didn't have a hard time coping with it. People cope differently with different traumas! Grief is different for different people. I have felt bad about it in the past, feeling like I'm not grieving enough. There is no such thing as enough. You feel what you feel, and there isn't a wrong way to feel after someone has died (or after any traumatic event!).
Not a beta but this book was proofread quite a lot from what I have heard XD just because you don't personally like or relate to the outcome doesn't mean it wasn't read over or was written badly. Totally valid to think it was anyway (I think Frugal Wizard had a major flaw), but others may read it differently!
There're more than two ways people can deal with grief. It's not just either a breakdown now or a breakdown later; sometimes there isn't that kind of breakdown ever, and that's valid and fine and realistic too. It might have been a punch in the face to you, or to me, but not to Sigzil. The breakdown might come after what we get to see, days later. Or it might never come.
Wanting to see that breakdown is understandable! I would have loved to see it too, actually, if there was one. But I disagree that us not seeing it was bad writing. It's one method of writing grief, and it is NOT an unrealistic one.
You could also consider the fear of Aux dying, that moment under the airships, to be the emotional climax, along with the moment of death itself. The rest isn't the focus. Which you're also allowed to not prefer as a style of storytelling, but it's a reading of the story that is valid.