Seems like the Safari Zone will be the greatest obstacle of the entire run. 50,000 people trying to coordinate walking to the Warden's Teeth in less than 500 steps... Infinite Monkeys don't even begin to cover it. It is bound to fail again and again, until they run out of money. That's when it gets really interesting.
As to not create an unwinnable situation, GameFreak wisely implemented a feature where the guard lets you into the Safari Zone at a discount if you can't afford the fee. The catch is... you get a reduced amount of Safari Balls. In the ultimate scenario, you get in for free, but only receive a single Safari Ball. Now, if that single Safari Ball is thrown, you'll be thrown out of the Zone...
My prediction is that this will remain an Internet phenomenom for months, where thousands (and later only hundreds) of trolls and a few devoted fans fail the Safari Zone game again and again. It will either fade into obscurity after a while, until it's either taken down, or the remaining hardcore fans manage to overcome the challenge, upon which the stream's popularity will spike again. That, or some mod will take control of the stream after a while and help the players through the Zone.
Either way, I can't see that Pidgeot remain in the party for long. It will eventually be the only decently-levelled 'mon on the team, and thus be subject to massive troll campaigns to release it.