I took it seriously, and it moved me.
Nope, I didn't laugh. Neither did people in the theater I was in.
Yes, it was an homage, taking from what has largely been considered the most popular Trek movie of all time. Regardless of the reasons why the writers choose it, I enjoyed it because it reflects both characters' arcs so well that I was in the moment with them, not thinking about TWOK, save for very surface level response.
Obviously, everyone will react differently, but I took it this way: NuSpock had a year of death. He lost Vulcan, and his mother. Then, he faced down death himself, only to be saved. Then Pike died. Kirk's death is the straw that broke the proverbial camel's back.
At that point I think Spock was going to snap. He was under intense strain, trying to reign in his emotions, and at every turn, there was someone dying, and all that experience of losing his mother, losing his planet, is back. At what point does that become too much to bear?