Different phones have different frame rate capabilities. My first-generation Pixel apparently can do the 240 fps referenced in the original article (I think it was posted in the Stearman Press subforum, just a couple weeks ago) -- at least, it claims 8x slo-mo and is nominally 30 fps HD in normal video mode. Also, Nyquist frequency is of less concern in this application, than counting frames in which the shutter was open vs. closed -- so you ought to be able to (approximately) time a shutter down to 1/150 or so at 180 fps and around 1/200 with 240 fps. Further, by recording multiple shutter cycles and averaging the counts, you can probably get pretty decent accuracy up to the full "one frame open" speed (in my case, about 1/250).
Sure, this still a limited method, but if you combine it with audio speed checkers (that record and detect the sound of the shutter opening and closing, for leaf shutters), you should be able to do pretty well without building or interfacing any kind of hardware detector. Honestly, for my large format operation, I don't care much about leaf shutter speeds above 1/200 or 1/250 anyway (relatively slow film, small apertures, my shutter speeds tend to run in the 1/25 range). And for my Speed Graphic, the focal plane shutter works correctly, including accurate curtain speeds, at least based on measurements made several years ago.