“But how can you justify a player character with a (non-disinherited) noble background in a dungeon-crawling fantasy game” well, the most obvious approach is a fantasy setting whose nobility practices cognatic primogeniture where, instead of “first son inherits, second son goes into the military, third son becomes a priest”, it’s “first son inherits, second son goes into the military, third son becomes an adventurer”. From the player’s perspective, it handily explains why the title comes with little material support from the family; from the family’s perspective, there’s an unspoken understanding that most of the spare heirs will be eaten by a dragon (or whatever), thereby simplifying the inheritance situation, and the few survivors will become great assets.
(There is, of course, the possibility that a surviving third son, having grown powerful and understandably harbouring some slight resentment, may return, kill his elder brothers with dark magic, and take over the dynasty, but in practice this almost never happens.)
… couldn’t Adventuring just be the fantasy equivalent of “Father send me to the War against the French to learn command and bravery”. Except it’s dungeons, magic and monsters.
Unfortunately not. One of the reasons it was traditional for the second son to go into the military is because it carried a warrior’s prestige without being especially dangerous in practice; any noble family of means would be able to purchase them an officer’s commission that would keep them off of the front lines, and you don’t actually want the second son to be in constant mortal peril, because you need a backup heir in case the eldest son dies unexpectedly. Being an adventurer – at least as depicted in most fantasy dungeon-crawler settings – carries too much real danger to play that role.
considering the third son is a priest and adventuring parties need healers who are often clerics