It's my honest opinion that 343 shoved the fans emotions on the Chief in Halo 4. Like it really came out of nowhere I feel, because honest they only spent about 3-4 months together, admittedly linked more closely than we can actually imagine, but it's not years like people think. Like he would be much more distraught over the loss of one of his Spartans I would feel.
I don't see how Chief's demeanor in Halo 4 is fan superimposition. Chief throughout the lore has been shown to care deeply for those he feels he's responsible for. They might have had a short time together, but it's what
they went through in that time that can truly matter. By Halo 3 (and this is under Bungie's subdued characterization) Chief was unquestionably confident in Cortana's plan, and risked life and limb to retrieve her in a hive of Flood. He even shared a touching moment when he finally reached her rather than just yanking her out of the podium.
Both we as the players and Chief as a character developed a bond with her throughout the various escapades of the original trilogy (and that's not even touching on the lore), so I found it perfectly fitting that Chief would be upset at Cortana's seeming demise.