They do exist. But because you’re not insta-killing them and they have, I assume, a 1 in 8,192 chance of spawning, they’re extremely rare. If you run an instance for too long, the game starts to slow down, or Pokémon may stop appearing altogether.
Speaking as someone who’s going for a shiny dex, I run the game eight times at once (four Chrome users with four incognito windows). By doing that, I can confidently say that within two to three weeks I’ll complete the shiny dex—except for the three dogs, due to their extremely low catch rate and rare random spawns. I’ve been trying to get them for four days now and have only encountered them six times as shinies so far.
You can also use a Golem or a Mewtwo Y to two- or one-shot Zapdos once you’ve gotten some decent prestige on them.