And all those other things I mentioned don't count, right? Have you ever considered that you might just be bad at the champion?
Anyways, I don't know how to do the whole separation by quotes thing, so I'll just go through them in the order than you had them.
Zyra has a snare, knock-up, and slow. Anivia has a stun and slow, Ori has knock up and slow, ziggs has slow and knock up, heimer has stun, azir's got slow and knock back, shaco's got fear, lux has slow and snare, cass has slow and stun. That's Zyra with three forms of CC and all the rest with two.
Less mana extensive than those three is a true statement. They might have lower costs, but they also have to spam to be effective. Ziggs and Ori have 2 second cds on their Q's, they're more mana hungry than Zyra for sure. Anivia's got the ult. I get by with Zyra building only morellos, can't really do that on any of those other three.
Two shots, not one. Which means you can prep it and have it block around 10 shots while still doing damage to it as well. That's pretty unrealistic though. Realistically, you can have it block 6 shots.
How so? Anivia's got a slower projectile than Zyra's E which you complain about being slow all the time. Her targeted damage is half effective when the enemy wasn't hit by Q or ult. AP Shaco's early game is heavy on mana, relies on a low base E damage and enemy team stupidity, trust me, zyra's is better. Ziggs again, slow ass projectiles and has high mana costs and low damage early on. And finally Cass, who either maxes Q for wave clear, but has less reliable damage, or E for damage and no wave clear. Regardless, Zyra should win this match up by having more range and wave clear.
Did you know that Zyra has a 1.75 second snare? WOW KNOWLEDGE IS GOOD. Also, during that charge time enemies inside take a huge chunk of damage and plants receive increased attack speed. It's more than just a knock up. Also, if you know it has a 2 second charge time, plan accordingly. Sometimes you have to prep a fight before it happens. If I've learned anything from Batman it's that every fight is winnable if you plan accordingly.