Hope I'm not to late to the party ;)

I played a lot of Anivia midlane in Season 4 and 5 but moved recently to the support role. I think Anivia can be a classic ap support like Brand or Annie.
So after playing something like 30 games Anivia support after 2 years, I accumulated some skill to pull an S rank frequently. So I will give my 50 cents to this guide ;)
Anivia is good in aggresive lanes and can be somewhat good in a stall lane (as always, there might be better supporters).
I would recommend to skill Q and E more equal because the lower Q cooldown can become handy.
I chose different items:
Coin, 3xpots and ward totem are my stadard start. Coin gives you some mana back and tons of gold. You are a poor supporter so take everything you can get.
After that I usually go for Boots and Sightstone or if the enemy laners were stompt, I try to rush Rod of Ages. I will discribe why later.
Now you can sell Coin and go for Eye of the Watchers for some more ap and wards (+ one free slot)
After that I usually go for Liandrys and Hourglass or Solari.

Supporters are the poorest champs in LoL. So you need to hit a pretty high gold efficency with your items. Also I don't think Tear is a good item, because Anivia doesn't have any cheap abilities to stack Tear fast like Ezreal...
Instead of penetration boots I would suggest to pick defesive boots like Ninja Tabis to get rid of the squshiness of Anivia.
I would definitly go for an early Rod because it gives you a perfect balance between hp, mana, and ap.
I haven't tried Hextexh GLP, but the freeze effect of GLP doesn't apply to your E so I thought it would be kind of useless and I don't think a dash is something Anivia benefits from.
I would never ever recommend a utility Anivia. To reduce Anivia to her W is not going to make you learn play her. Go the aggressiv build and watch one or two Anivia midlane videos.

I know you think the TP support is useful and you can win games in Bronze (no offense) but think about it this way:

-Enemy ADC: Flash Heal + Enemy support: Flash Exhaust/ Ignite
-Your ADC: Flash Heal + You support: Flash Teleport

Enemy team has 4 summoner spells that can be used in fights. Your team has 3 summoner spells that can be used in fights. 4>3. You will most likely lose in a pre lvl 6 fight.

Now about Anivia support:
You have an AOE stunt, a Terrain ability and a nuke (if frozen). You unlock your main combo lvl 2 with Q+E (stunt+nuke). At lvl 3 you add a displacer. Anivia base damage is good, you can get a cheesy kill lvl 2-3 with Ignite (same as done with Annie/ Brand/ Trundle/ Vel'Koz).
Without Ignite, enemy ADC Heal is 100%, so add the extra 90 HP lvl 1. Add also the Exhaust, your damage cut by 40%. You wont be able to kill none.

Your items (mana scaling like Rod of Ages and Tear of the Goddess) doesnt add any damage in lane. Enemy will build AD and AP, they do more damage.

Summing up: Get Ignite so you can get lvl 2 kill. Build AP soon, the Morellonomicon+ Frostfang will help you with mana. Even upgrading to Frost Queen's Claim to get the slow active to land Q easier will be good.
The idea is good, but you need to improve the build and summoner spells. Trust me, you will win more with this items. At least try it out one game, you wont lose anything. Good luck.