It depends on the playstyle you excel at and what match up you're against. I look at Zyra the same as Morgana, which is a great support.
I played a good deal of Tahm support and this is how I felt. He takes a few levels to really get useful. His poke can be decent to harass, but I'd hardly call it zoning. His devour can be good to get in or out of an engagement, but adc's rarely enjoy being moved forcefully, even if it IS the right choice. It can work either way. His shield is really just for absorbing something like Lucian's ult and either regenning a lot back or popping a shield to survive it. His stacks can get pretty oppressive if the enemy let's you stack them. I personally don't LOVE him as support because he doesn't have the insane survivability until later in the game and you need his ult's passive to really do dmg. His harass is hampered by minion blocking and being too slow to reposition with no other harass. Examples I put him against are Sona, who's q is legendarily oppressive with long range and no minion blocking, morgana who can have her q blocked but her aoe field can zone and harass as well to force them into the open, or blitzcrank's W helping him reposition quickly to land a q. Tahm can't zone without a clear shot, he can't get around because he's slow and telegraph's and he doesn't have any range except for his Q. However, that all said, he isn't the worst support out there and his late game AND his ult can be great.
Zyra can be played to zone really well. Her poke is great, she has the CC from her Q which always scared the shit out of me when I was adc, her plants are annoying at best though. Her ult can really fuck your day when you're getting ganked or about to get dove. She has low health because she was never meant to get hit. The idea is to keep people at bay and never let them be in a position to actually retaliate. She doesn't do much to protect the adc in that sense, but she can certainly help secure a good kill shot with her CC and damage. They are actually reworking her to give her more of a support role as well. I shudder to think of how powerful she may be after the mage update...
Between the 2, I'd grab Zyra unless you intend to also work top lane. Tahm CAN be support, but he's certainly more of a top laner than a support. I see zyra in support more often than Mid like she was meant to be played, but after the mage update she'll be even better as a support.
I dare say look at Cassiopia as well. Her rework may just make her a good support. I have been watching it carefully thinking up ways to fuck people's worlds with her when the time comes.