There are a lot of factors as to how KAAY and other 50 kW and other high power directional stations can be stronger than local signals. Directional Antennas are a key factor in the highest skywave returns in the US, where 50 kW is the limit. Close in to the nondirectional Class As, the skywave can be on the order of 3 mV/m. Directional Class As can be as high as 10 mV/m maximum. So you have to compare that with the actual IDFs in nulls and minor lobes of local stations, along with their NIFs. Even with fairly well protected NIF contours, and especially in nulls, the signal is less and has more interference than the Class As did, and should still.
A college acquaintance of mine used to hang out with the Milner family (who later owned WSRF/WSHE Ft. Lauderdale, and who now own a cluster of stations in Kankakee, and did own the 99.9 which "moved in" to the Chicago Market in Park Forest, IL). The older Gene Milner owned WTAC 600 at the time, which had their towers North of Grand Blanc. The Milners lived in Grand Blanc, MI, in a subdivision about a mile Southeast of the towers in a deep null, with a lot of phase distortion. My college acquaintance would ride around with the Milners at Night, and would tune in to WCFL instead of Gene Milner's station, because it came in so much better! So that may be where the family first became interested in Chicago Area radio. A station engineer once told me that station owners and managers somehow all end up living right in the deepest part of a null, and are always complaining about not being able to hear their signal well!
It never sits well when the engineer tells the owner that if he wants to hear his station better, he'll have to move! I have known of station owners who would literally try to adjust the phasor themselves to try to get a better signal at their house! Not the Milners to my knowledge, though they did have to redesign the DA at WSRF according to another engineer.