I see why and the dark backstory trope is funny to grating sometimes depending on the level of edge the player inserts, but on the other hand I always think: Why would a happy character with an intact social network and support system and a loving family... why would they go on deadly adventures? If they are happy, would they not stay in their happy life with their family, farming and enjoying life?
While this is a question worth asking, it's not a particularly difficult one to answer.
Perhaps the character just felt called to do it--maybe they saved someone's life and now feel that, since they
can save lives, they
need to do so. Great power and great responsibility. Or, for a religious character, maybe their faith motivates them to it. There are a dozen gods in Faerun alone who would be likely to have worshipers with healthy family lives with living relatives, and yet who would be likely to encourage their worshipers to set aside creature comforts and help make the world better.
Perhaps the character is intensely curious, and just needs to
know--the adventure is secondary to the knowledge and experience. I could see plenty of spellcasters taking this path, and not just Wizards either. Gleaning the nuggets of true enlightenment or perfecting a true masterpiece can be powerful motivators even when someone has a loving, happy home.
Perhaps something has put that family under threat, and it's
for them that the character is adventuring. Someone has grown ill, maybe. Or someone went missing, and the character has taken it upon themselves to find out what happened (that one actually was a character's backstory in Hussar's LMoP/PaB campaign). Heck, it could be that someone
came back, injured, and asked the character to take up the mantle on their behalf.
Perhaps the character has gotten
bored with their safe, cozy life. They want to feel risk and danger--something they've never really had to face, because they've always had a robust support network, stable family finances, and caring family members
always present, always right there. Getting away from their family so they can actually experience the adventure they want. It might end up being too much for them, or it might galvanize them because they can draw on that solid foundation.
Perhaps a relative of theirs was an adventurer, and they're inspired to follow in their footsteps. Doubly works if said relative earned a great deal of money from adventuring, and that seed is what
allowed the family to live comfortably and focus on building one another up. Building up a tradition, like how Bilbo's adventuresome past helped give Frodo a good home after his parents' death, and was part of the inspiration for the four hobbits going off to join the Fellowship.
Point being: There are a lot of reasons why even someone from a good home, with a loving family,
overall stability, and a lack of Horrible Awfulness in their past, could still want (or need) to adventure.