> [Interacts] only through the weak nuclear force and gravity, or possibly other interactions with cross-sections no higher than the weak scale 
But Axions interact via the electromagnetic force which is stronger than the weak nuclear force 
First: If the axion exists, its electromagnetic coupling is extremely small.
The axion-photon conversion would be a Primakoff process, which can proceed through a loop. Loops suppress cross-sections drastically. The axion is not electrically charged -- we would know all about them by now (and they could not be the dark matter).
If the axion is the mechanism by which the CP problem is resolved and the dark matter is composed of axions, then there are predictions for the strength of the effective axion-photon coupling constant. What is special about the ADMX experiment, and why Quanta is writing about them, is that ADMX is beginning to explore that region in parameter-space in a definitive manner. If both of those conditions are met, there are now masses which the axion cannot have, or ADMX would have seen them.
Second: The weak force is not, in many senses, radically weaker than the electromagnetic force. We don't experience it in our daily lives because the force-mediators (the W and Z) are massive, while the photon is massless. Massive force-mediators severely restrict the distance over which a force can act. The coupling constants of the known forces "run" with energy, and appear to roughly match at the "GUT" scale: https://en.wikipedia.org/wiki/Grand_unification_energy