There are pros and cons to either way- the baseball way or the softball way.
You say you prefer the baseball way. But what about this:
Close play at the plate, catcher misses the tag and runner misses the plate. Since the standard on
all other tag plays is that it must be signaled as either "safe" or "out", if you make the single exception on this play to "do nothing" (give no signal) it really can tip off the defense that the base was missed.
Even if they didn't see the missed base, if the umpire "does nothing" you can almost always expect an appeal to follow.
You might as well just have the umpire announce out loud, "He missed the base"!
Of course, it can also tip off the runner that the base was missed so he needs to try and touch it. By rule, if he doesn't touch it, then walks off the field and the defense doesn't appeal it, he's still safe and the run still counts.
To me, that tips the scale of "fairness" just a little bit more in the defense's favor. It can alert them to an appeal, and an out, that they may not have otherwise been aware of on their own.
It could also be argued that giving a "safe" signal might lead a runner to
assume that she is now and forever "safe", no matter what happens next. But if the players understand the rules, they should know that isn't always the case.
Since I work both sports, I've had plenty of chances to see teams get confused on this play, no matter what the umpire did or didn't signal! The bottom line is that no matter what the umpire does, the ultimate responsibility to know if a runner touches a base or not should lie with
the players playing the game.
I think that is why I prefer the softball mechanic for handling this. Since "safe" or "out" is signalled
on every other tag play, an umpire giving a "safe" signal here is more of a normal expectation. It shouldn't tip off either the offense or the defense that the base was missed. It leaves the final responsibility of retouching the base or making the appeal up to
the players.