Runner just has to complete his base running assignments. As an umpire, I'll be saying, "Runner is awarded third base" and not adding "once he tags up because he left first early". It's incumbent on the runner to know their job and it's incumbent on the defense to know their job.
But yes, on a dead ball award, all bases must be run legally and all bases must be touched (exception being over the fence home runs in adult slow pitch USA where the ball is dead, all runners are awarded home and no appeals are granted based on a runner missing a base). USA Rule 8.3. B, H, and I specifically talk about awarded bases.
Also 8.5.G specifically talks about this situation with an overthrown ball and a runner completing their base running responsibilities.