Perhaps this is just the consequence of how I study. I learned 息を吐く as a set, and if you search dictionaries for the string 息を吐く you will get entries for いきをつく in my experience. And seeing as つく is a Kanken pre-1 reading for 吐く while はく is jouyou, I’ve just got it at the forefront of my mind right now. Such dictionaries will also include the physical meaning of just letting out a breath as well. Dictionaries will not tell you how often the physical breath meaning is used (that is to say, hardly ever apparently).
So, I suppose I failed to keep in mind that 息をつく was an expression that could co-exist with the “non-expression” / “just the words at face value” string of 息をはく. Essentially, there’s no reason for there to be dictionary entries for 息をはく because it’s just what the words mean with nothing more, and with both kanji being not particularly advanced as jouyou kanji, I haven’t encountered an いきをはく sentence in my kanji studies in a long time (if I ever did).
This is the kind of thing that shows up on Kanken pre-1 and is supposed to catch you out for thinking it’s はく where only つく is right. Or you will be presented with the reading of つく and have to write 吐く. However, obviously つく is not always right, and I lost sight of that.
