Did Athletics die out after Ben Johnson's positive? There's been drug issues in athletics for years before and since.
Wiggins is a big star now, but he's still a much bigger star in Britain than anywhere else. And our hypothetical Sky story is kind of predicated on it happening NOW, when the Armstrong backlash is in full effect. A few months down the line, that will have died down and though the sport will be up the creek in the UK and maybe the US, knee-jerk reactions will not be as extreme.
The sport is not 'safe', so to say, but that's more because the UCI has staked a lot on globalisation into markets that are not ingrained cycling cultures and are therefore more vulnerable to backlash like that. FDJ, Cofidis, Euskaltel, Lotto and Lampre have weathered a lot of pretty heavy storms in their long time in the sport. Rabobank may have got the hell out of Dodge, but they probably knew something was going to go on into their old team that led them to choosing the right time to exit. If Cofidis can escape multiple soigneurs and riders getting arrested with doping products, and team members publicly calling every team member bar 2 dopers, right in the middle of the Armstrong era, and then have the whole team withdraw from the 2007 Tour - their home race lest we forget - in the wake of positive tests, and still be in the sport 6 years later, why the hell would some guy who rode for them years ago testing positive while riding for a completely different team make them go, "you know what, this cycling is a bad game. All that drugs. Let's get out of it"?
Also, I hate to break it to you, but cycling's still not that big a sport. Usain Bolt testing positive would have more of an effect on athletics than Bradley Wiggins testing positive would have on cycling, and athletics can survive Bolt.