I interpret that "may" as being "empowered." If the umpire determines the situation warrants being rectified and there is a fair remedy, they have a responsibility to do it.
Therein lies the rub...if the umpire doesn't determine that the situation warrants being rectified, then he does nothing.
That determination is where the "judgment and discretion" that I mentioned comes into play. A coach might not agree with his judgment, and sometimes his judgment might be poor judgment, but his judgment is his judgment- it is what it is.
This one got me digging a little deeper and I found some interesting rulings. The high school case book has no fewer than three different sample plays involving reversed ball/strike calls on full count, each with different twists as to the runner getting tagged out or not, or if she was stealing on the pitch, or the if batter getting thrown out at first. Each one has a different twist in the ruling depending on the circumstances surrounding the play. So, again, there is no "one size fits all" solution to rectifying this play.
Unfortunately, there isn't a case play that exactly fits the play we're discussing, where R1 safely reaches second base and the catcher is late in her checked swing appeal.