In the 1967, 1970, 1974 and 1977 rulebooks, per 10.2 mate only occurs if check cannot be parried and the definition of "parried" was not given. Thus an illegal move delivering check is easily parried by citing the illegal move, which must then be retracted and replaced with a legal move.
If the Bachler-Dowd game occurred at that time then the TD was blindsided by a carefully prepared unusual parsing of rules and apparent logic for which the TD had not yet had enough time to find a correct rebuttal. It could have been readily denied under the rules as then written.
In the 1987 rulebook the requirement that the mating move be legal was added to 1.12A, so the Bachler-Dowd game must have occurred prior to that. The rule about mate immediately ending the game was also added to 1.12A at that time. That rulebook also described HOW to parry a check in 1.11A and did not include citing an illegal move as one of the methods.
The 1994, 2003 and later rulebooks use a different rules numbering sequence, with 12A defining check. The 2003 and later editions include the TD tip in 11A that can handle an illegal move shortly prior to a mate with a legal move.
That would mean that the 1987 and 1994 rulebooks might have bowed to pressure to more fully define getting out of check, and in doing so may have opened a door for the type of tortuous logic used before "parried" was given a limited definition. The door was partially closed with the "legal move" addendum for the mating move but it wasn't fully closed until the 11A TD tip was added (no later than 2003).
The 1987-2003 window is an example of how trying to more fully define every nuance in a way that closes the door to misinterpretations often ends up opening new misinterpretations that are harder to refute after the revisions were made. The rule was fine with using "parried" without a further definition. I am guessing it was added to explicitly refute people trying to say that a king couldn't capture a checking piece (or one of the two pieces giving double-check). The immediate end of the game was likely added to state that a flag could not be called after a checkmating move was determined, though it reinforced 18G2 so that a TD could justify calling a checkmate (or stalemate in that rule) because there was no longer an active game to be interfered with. It might have also been added to avoid people trying to annul a checkmate by delivering a counter-checkmate.
It may have been better to
A> make 12A simpler, possibly also copying in the final sentence of 12C and eliminating the rest.
12A. Definition. The king is “in check” when the square it occupies is attacked by one or more of the opponent’s pieces; such pieces are said to be checking the king. Check is parried (a player gets out of check) if the player’s next legal move ends with no piece of the opponent’s attacking the square the king is on. The king cannot parry check by castling.
B> not say that checkmate immediately ends the game (thus allowing illegal moves to be cited)
C> leave the mate superseding flag rules only in the sections dealing with the clock
D> let TDs use just 18G2 if they need to intervene after an unnoticed checkmate/stalemate
E> possibly also copy the TD tip (about not gaining an unfair advantage from an illegal move) more cleanly into 11A