A batter overruns first base, but misses the bag. Is then tagged out. Does the runner on 3rd score?

Lifting much of my answer heavily from Wax Eagle's answer here:

From MLB rule 5.09(c)(2)

[The runner is out on appeal when] With the ball in play, while advancing or returning to a base, he fails to touch each base in order before he, or a missed base, is tagged.

This basically means that a runner is considered safe after they've rounded first base without touching it, that is until either he or the base he missed is tagged before he returns to it and the fielding team appeals to the umpire.

However, this specific scenario is addressed in rule 5.09(d)

Unless two are out, the status of a following runner is not affected by a preceding runner’s failure to touch or retouch a base. If, upon appeal, the preceding runner is the third out, no runners following him shall score. If such third out is the result of a force play, neither preceding nor following runners shall score.

I read this as meaning that a "missed base out" (i.e. missed the base and then tagged out) can still be effectively counted as a force play out; being the third out of the inning, no runner can score. In the situation you describe, the umpire might have technically needed to have been appealed to, but in practice there often isn't an official verbal appeal--the umpire knows the deal.

Although the batter/runner would have been safe if he touched the bag and although he was considered safe for a few moments before getting tagged, he ultimately did not safely reach first base to avoid the force out.

This is actually addressed pretty specifically in a comment in the 2016 version of the Official Baseball Rules pdf.

The comment to Rule 5.08 (pages 37-38) has this approved ruling:

APPROVED RULING: No run shall score during a play in which the third out is made by the batter-runner before he touches first base. Example: One out, Jones on second, Smith on first. The batter, Brown, hits safely. Jones scores. Smith is out on the throw to the plate. Two outs. But Brown missed first base. The ball is thrown to first, an appeal is made, and Brown is out. Three outs. Since Jones crossed the plate during a play in which the third out was made by the batter-runner before he touched first base, Jones’ run does not count.

and this approved ruling:

APPROVED RULING: Two out, bases full, batter hits home run over fence. Batter, on appeal, is declared out for missing first base. Three outs. No run counts

In both cases, the batter-runner is the third out on the play for missing first base. No runs score.

The umpire was correct. Batter /Runner is assumed to have touched first base when he passes by it even if he missed it. When the throw comes late he is not automatically out just because he missed the base. Defensive team must appeal to umpire before batter/runner comes back to touch the base. Then with the ball step on first or tag batter/runner. "Hey ump he missed the bag." If on appeal he is called out the run does not count.