At first, it may seem like the character's behavior is not something a person would actually do, an important thing to consider is the 5e interpretation of HP.
Hit points represent a combination of physical and mental durability, the will to live, and luck. Creatures with more hit points are more difficult to kill. Those with fewer hit points are more fragile.
A creature’s current hit points (usually just called hit points) can be any number from the creature’s hit point maximum down to 0. This number changes frequently as a creature takes damage or receives healing.
Whenever a creature takes damage, that damage is subtracted from its hit points. The loss of hit points has no effect on a creature’s capabilities until the creature drops to 0 hit points.
If the hero gets hit by a spear trap and fails the Dexterity save, that doesn't mean the trap draws blood. He may narrowly dodge it and get more tired, or have it be absorbed by the armor, leaving a bruise. This wouldn't necessarily deter a headstrong hero as much as, say, getting stabbed in the gut, so his behavior is reasonable (for a fantasy hero).
A character won't be visibly wounded until multiple attacks. At my table, I've house-ruled that a character is "bloodied" (ala 4th Edition) at half-health, and I tell my players when an enemy is bleeding.
If this sounds like something you would like, you could add this to your game and also rule that a character bloodied by a trap must succeed on a Charisma save or become too frightened to walk in front anymore.
