There is just no downside to Zyra in her current state. She has CC. She has tons of range. She has really hard burst. She has sustained damage and zone control. She doesn't really have mana issues. It is absurd that she exists.
Did you play many Zyra games? I may be the worst player in the world (I am) and that is the reason I nearly always did terrible with Zyra, but to say she had no downside is absurd.
Kat and Ahri were already examples of champions with good mobility who can dodge and assassinate. I think I already mentioned them, and they are specifically called out as even better against Zyra now. Problems with Zyra can include:
1. She is slow with no blink or gap closer,
and even slower after the nerf.
2. Combined with 1, she has no escape mechanism besides flash, attempting to snare a ganking champion, or hoping that the ult knockup will slow a champion long enough for them not to gap close, flash, stun, etc. to kill.
Note that this is worse after the nerf(s) since enemies can more easily escape the ult.
3. Against manaless champions or heavy pushing champions that can easily kill most of a minion wave with 1 ability (Sion, Galio, Ahri, Morde may fall in this category), Zyra tends to need two abilities to clear a wave. Attempting to push quickly or defend such a push using two abilities means you are constantly losing 170+ mana, out of a total of 500-600 mana early. Doing that twice means you are in danger of not being able to use more than one ability due to being low on mana.
Note that this is worse after the nerf since base damages have been lowered.
4. If Zyra does push a lane heavily, she becomes very vulnerable to ganks due to note 2 above. As I believe has been noted by a GAF poster before, ganking her repeatedly can be a good strategy to dominate her. Your only options if ganked are to try to snare/ult/flash to escape, or to fight and hope you can do enough damage to survive one of the ganking champions.
Note that this is worse after the nerf since base damages have been lowered, and the ult is easier to escape. Seeds can be used to ward parts of the midlane bushes, but this leaves you vulnerable and with fewer damage and combo options due to no/fewer plants.
5. Unless the Zyra player is great at last hitting micromanagement, creating plants in range of enemy minions can lose lots of last hits (plant nearly kills a minion causing you to miss a last hit). I am not sure I have ever seen a Zyra outfarm a midlane opponent in the first 20-30 minutes of a game, though presumably this is because I have only watched the worst players in the world use her. Using more abilities can mitigate this, but as mentioned in note 3, mana reserves will suffer.
6. The snare is difficult to hit unless you get in range for burst mages to destroy you. Note that this is probably because I (and other Zyras) are the worst players in the world, and good players hit it all the time. I suspect that Zyra may only be overpowering if her snare hits most of the time.
7. My understanding is that her ratios are mediocre, aside from the ultimate. Riot suggests building glass cannon on her may be an option in the future, but I really do not see how she would be able to do enough damage that way compared to other burst/DPS casters, especially with reduced Q range and mana issues if she attempts to harass or combo often.
In short, I feel that a lot of people complain about her either without playing her and dominating consistently (thus helping prove their point), or without learning her weaknesses and how to counter them. In particular, if Zyra misses a snare or it is dodged, and/or if seeds have already been used, champions who can gap close and/or stun Zyra can do heavy damage to her without being snared or having additional plants put on them.