Even if we are locked in as an ex-soldier, we can define ourselves as a sniper, infantry, pilot, cybernetics, or something else. So if we were a sniper in the army, then we can play as a sniper for the rest of the game. The reason why we have to relearn our sniper skills could be due to whatever process that was used to skip 200 years. The wife seems to have even more freedom since she could have been a mechanic or chinese spy for all we know. You can play the game however you want.
Unless finding your wife and child is the main quest, then you can treat the tutorial as a dream sequence caused by cryosleep or whatever it was so there is no wife and child. The tutorial or even the rest of the game could be Total Recall for all we know. Would actually make for an interesting ending. Protagonist wakes up back in 2078 from a VR pod to a world where the bombs never dropped and they were just playing the latest VR game.
The Fallout games has always gone with the set protagonist. Fallout 1, 3, and 4 are Vault Dwellers. Fallout 2 is a tribal. Fallout Tactics and Fallout New Vegas are the only ones where the history of your character is not revealed. Fallout Tactics is some Recruit while Fallout New Vegas is a Courier where we don't know where they grew up or what they did before they got shot in the head except for the history revealed in the Lonesome Road, but that is only one minor part of the history. It seems like most RPGs follow the set protagonist.