My problem was that I was conflating these two events into simultaneity -- "The pilot waits until a month elapses, and calculates what the time is on Earth. Only five days have passed; it's January 6th. Then he slams on the brakes, turns it around, and heads for home ..."
Quite obviously, a lot happens between the point at which he satisfies himself that he has gone far enough, and the point at which he actually turns around. A lot of "events" get telescoped into that interval.
Precisely. The solution to the twins problem is sometimes referred to as the "simultaneity gap", but I consider that a misnomer, because there isn't an actual gap, unless you contrive something like my acceleration-free example. But during the turn-around, there's a hell of a "fast-forward" that takes place back on Earth, from the traveller's point of view.