There is no requirement that it be a "kicking situation" in the rule book. Further, there is no requirement that you actually kick the ball. A scrimmage kick formation is as follows:
"ART. 2 ... A scrimmage kick formation is one in which no player is in position to receive a hand-to-hand snap from between the snapper’s legs, and at the snap, either:
a. A player is in position with a knee on the ground 7 yards or more behind the line of scrimmage, in position to be the holder and receive the long snap and with another player 3 yards or less behind that player in position to attempt a place kick, OR
b. A player is 10 yards or more behind the line of scrimmage and in position to receive the long snap."
You wouldn't be able to trot your punt team out on 1st - 3rd down according to rule 7-2-5b (Exception), unless you have the required at least 4 ineligible numbers. Your PAT/FG team almost certainly will have the required numbers, but if it didn't, you'd have to sub in the required amount of ineligibles (at least 4 in a scrimmage kick on 1st-3rd downs).
It was put in place to get rid of the possibility of the A-11, but it restricts something it wasn't meant to restrict, which sucks. Also, just from experience, I doubt too many officials call you on it, because many (not most, but many) are barely (if at all) aware of the exception, or it just doesn't occur to them in the moment...particularly among those who primarily officiate small class football, but not always limited to them.