Then on fly balls or line drives that are caught and when the runner didn't tag up, why is it that you don't have to say "The runner didn't tag up??" Since that is a live ball appeal play??
Using the phrase that the runner missed the base is used in a dead ball appeal.
There is a concept taught to umpires that any appeal must be an
unmistakable appeal. The defense needs to make it clear to the umpire just what exactly it is they're appealing.
When a line drive or fly ball is caught and a runner leaves his base too soon, the defense making an immediate throw and tag of the base is considered to be an obvious (ie: unmistakable) appeal. Thus, no verbal appeal is needed.
If the runner left early and there were intervening plays on that, or other runners, a throw to tag the base left too soon at might not be an obvious appeal. The umpire needs to know what exactly is being appealed and which runner is being appealed. A verbal statement may be required to make the intent obvious (such as, "#14 left early on the catch").
Another concept is that
accidental appeals are not to be ruled upon.
Suppose a runner left second base early on a catch by an outfielder. No immediate appeal is made, there are other throws made and the ball eventually winds up in the shortstop's possession. The SS is walking the ball toward another runner off his base and just happens to accidently brush his foot across second base.
The appeal is not obvious and a verbal statement would need to be made to make the appeal valid.
First base is treated a little bit differently than other bases due to the nature of plays at that base. Runners may overrun the base and return without liability to be put out, so long as no attempt toward second base is made.
When the batter-runner beats the throw, but misses the base, the defense needs to make it obvious they're appealling the missed base before the runner returns and touches it. The verbal appeal serves to make it obvious.
Without the verbal element of this appeal, the defense doesn't make it clear that they even saw the missed base. The ball is being thrown/fielded to first as part of the normal play, so the fact that the defense has the ball and is touching the base isn't an obvious, unmistakable appeal. In this case, it falls more under the category of an "accidental" appeal.
Simply tagging the runner isn't sufficient because it wouldn't be clear if the defense is attempting to tag the runner because they think he tried to advance to second base or as part of a missed base appeal. So, a tag alone isn't considered unmistakable either.
You might not find any of these concepts in the rule book, but they do appear in umpire manuals and they are taught in advanced classes and training. But this play is one that is well documented in training literature and this is how umpires are taught to handle it.