This is a good question. And I agree with your instincts that the reader won't want to dwell on the grief because they won't care. The reader wants the character to get over their grief and get on with the story, but they need an excuse to do so (otherwise the MC seems emotionless and weird).
I think some time manipulation can do a good job of this. For example, 'The main character spent a full month in a profound state of grief. The memory of his dead family was intense and painful, but over the weeks, began to fade. It didn't fade all the way, and it likely never would, but it faded enough for him to get on with the rest of the story.'
But what if the family death happens during the inciting incident and we can't just skip over a month of grief in 3 sentences?
This is a very interesting statement because in some ways its true and in some ways its not.
You say that the reader wants the character to get over the grief and move on and that sounds really cold. Detached, cold, cruel and harsh on the character. Yet its actually often what readers want - we want to see characters change and react to events, but we also want them to often be heroes or at least to rise to challenges they meet.
A character who doesn't, or who dwells on their failures too long is actually super hard to create, especially if the novel focuses on a single character point of view for the story (with multi-character stories like Game of Thrones and such, you've a lot more scope for a depressed character because they are balanced out by other characters). Robin Hobb I think encountered this with her Soldier's Son series where the lead character spends much of the story almost being swept along by the world around him and spends a lot of the 2nd book in a very depressed state. I find it very interesting because many find the story to not be one of her best, and yet I think she actually took on the challenge and achieved it really well in writing a character who, fundamentally, isn't much of a hero and who is rather depressed and crushed by the weight of many aspects of the world and events which happen to them.
As for emotional trauma early in a story, look at Disney and Pixar. They use early trauma a LOT in their story structure. Consider a story like UP or Frozen or Anastasia. In each of those stories the lead character experiences serious trauma/loss of a loved one very early in the story. In stories like UP its quite an emotional event for character and, honestly, viewer. In Frozen and Anastasia the event isn't glossed over, but it is sped past much more so. It becomes an event which shapes the characters, but we don't dwell on the immediate traumatic period after, but rather after a time advance.
The other approach is the Lion King style, where the trauma is much later and holds much greater impact on the audience. Interestingly even with that greater emotional contact they still use a time advance, though that's more for the practicality of aging the character up to fit the way the story evolves.