Even if you successfully predict the Azumarill switch a 96.1% chance of having to eat a Play Rough anyway is unpalatable to say the least. I'm not sure making for an extremely shaky check to Togekiss, Azumarill and Clefable warrants running Flash Cannon on Hydreigon. I notice you've left out Sylveon, no doubt because it makes your argument look a lot worse.
252 SpA Choice Specs Hydreigon Flash Cannon vs. 252 HP / 252+ SpD Sylveon: 154-182 (39 - 46.1%) -- guaranteed 3HKO after Leftovers recovery
4 SpA Pixilate Sylveon Hyper Voice vs. 0 HP / 0 SpD Hydreigon: 600-708 (184.6 - 217.8%) -- guaranteed OHKO
Hydreigon really, really does not have any business staying in on anything with a Fairy typing not called Mawile (or Klefki, but that's UU now). Flash Cannon is an alluring trap.
Well he only said some fairies, not all, but from the calc you provided it looks like Sylveon get mauled about the same as the others anyway.
252+ SpA Choice Specs Hydreigon Flash Cannon vs. 252 HP / 4 SpD Mega Mawile: 145-171 (47.6 - 56.2%) -- 84.4% chance to 2HKO
252+ SpA Choice Specs Hydreigon Flash Cannon vs. 252 HP / 252+ SpD Florges: 148-176 (41.1 - 48.8%) -- guaranteed 3HKO after Leftovers recovery
252+ SpA Choice Specs Hydreigon Flash Cannon vs. 252 HP / 0 SpD Klefki: 157-185 (49.3 - 58.1%) -- 64.5% chance to 2HKO after Leftovers recovery
Thats about every relevant fairy I can think of, and they all take about 40% minimum from flash cannon.
Naturally you don't stay on anything unless you can KO or are foddering, but those calcs mean that those pokemon can check Hydreigon only a few times. Since all of those pokemon are slower than Hydreigon, it means they can only stop the beastie once before they can no longer switch in.
The advantage to running Flash Cannon isn't so that Hydreigon can check fairies, its so that fairies can't check Hydreigon. This is why it was called an uncounterable pokemon last gen, as nearly everything was 1 or 2hko'd, and its counters could only safely switch in a few times.
I haven't played around with many choice'd variants of Hydra but if you were to run Flash Cannon the biggest loss would be the coverage. Hydra already wants to run Draco Meteor, Fire Blast, Dark Pulse, Roost and Superpower before you consider a Steel move. I haven't tested Flash Cannon Hydra myself but I don't see how it couldn't be at least worth testing.