Jynx is really frail though. It can cause Zapdos problems, but Zapdos itself outspeeds, does a shit-ton of damage with Thunder, and can paralyze with it, too. Plus, Zapdos can always RestTalk off an Ice Beam, whereas Jynx is stuck with that Thunder. I'd honestly switch Jynx out of that lead matchup most of the time. Jynx I see as better for keeping Zapdos from switching in so that you get a better sleep, not for going toe-to-toe with Zapdos in an equal-turn head-to-head.
Jynx is probably best used as a lead, though, because otherwise, how often is it going to have a chance to be active? I'd just keep in mind that a Jynx lead is mostly good for forcing Zapdos to either Thunder or switch (that's all it has leeway to do), and not necessarily for auto-forcing Zapdos to switch out. Regardless, Jynx is a pretty gimmicky option, and considering how little it adds to team defensive synergy, there's usually better options out there.