Having been in the military, I can assure you that they do not train you to be unemotional when under stress. They don't want your emotions too affect your thinking but they aren't really interested in having everyone act like Mr. Spock.
Most people when they suffer one psychological shock after another will go into a kind of autopilot. They will get the Infinity Gaze and pretty much only react to their surroundings without much thought, while their brain tries to put some semblance of order to the world.
We don't have all the details, but basically the protagonist has been experiencing many life altering shocks over a very short period of time (at least from his/her viewpoint it is a short period of time). I don't know if it is intended in the game, but in real life the effect would be pretty similar. The protagonist wakes up and starts the normal daily routine. Bathroom, gets dressed, check on the kid, cornflakes and OJ, etc. Suddenly the sirens go off. Grab the kid and run to the Vault. Watch the nuke hit as you are lowered into Vault 111. Get processed in and stuffed into cold storage. Wake up from that and find the whole Vault dead. (It is unknown at this time if you find the remains of your spouse and baby, either way, they are no longer with you.) Exit the Vault and see the world you once knew has been replaced by a terrible wasteland populated by nightmarish beasts and other abominations. From the protagonists perspective this all happened in the space of a day or two.
The protagonist is acting like a dullard? It is really a wonder that some feral ghoul didn't find him/her laughing hysterically outside the Vault entrance and eat lunch.