Just would like to chip in and compare Zyra to another mid-laner who also is a popular support, who also is immobile, has a game changing ultimate, and AoE CC and damage.
Zyra's Q: 210 + 65% AP (5 Second CD)
Zyra's E: 200 + 50% AP (12 Second CD)
Zyra's R: 350 + 70% AP (1 Minute 10 Second CD)
vs
Annie Q: 220 + 80% AP (4 Second CD)
Annie W: 250 + 85% AP (8 Second CD)
Annie R: 425 + 80% AP (80 Second CD)
Both of these are Mid Laners who are extremely immobile, have AoE Crowd Control and damage, are squishy, and mana-hungry with big game changing ultimates. Both of them also are viable and popular supports.
Annie however, while her Q isn't reliable is point and click, has higher base damage on everything, and has higher AP scaling on everything while instead of just having a Snares and Knock-Ups, all of her abilities except her E have the potential to stun people for up to 1.75 seconds. The longest duration of Zyra's E snare is 1.75 seconds (It's a snare, not a stun.) and the knock-up on her ultimate is 1 second at all ranks.
These two champions aren't a whole lot different in terms of their roles and playstyles. They're both Mages who are good Supports, that manipulate Crowd Control and have scary good Ultimates. Despite this, Annie is more reliable, has higher scaling, has higher base damage, and also has noticeably lower cooldowns.
Literally the only reason to pick Zyra over Annie would be due to Zyra's fairly long range. Despite this, all of her abilities are not instant and her Q, E, and R all have fairly longer travel times/delays or whatever they're called. To be honest, there's not a whole lot of reason to be hit by Zyra a lot. Her Q has a 1.5 second delay if I remember, her E has a fair Travel Time however it's a VERY necessary part to her combo and even if you're super fed, you're going to need more than just your Q to kill someone, and finally her ultimate has a 2 second delay which is decent for how large the radius and range is, though people can quickly move out of it if not CC'ed.
tl;dr Zyra is inferior in most aspects to similar CC Burst Mages/Supports including Damage, Scaling, Cooldowns, and Reliability with basically her only strong side being Range.