Blitz Q-R : (190 + 1AP) + (250 + 1Ap) totalling 440 + 2*AP.
Annie R-W-Q : (150 + 0.8AP) + (70 + 0.8 * AP) + (175 + 0.8 AP) totalling 395 + 2.4AP
So Annie's base damage is 45 lower.
This page shows that the most common runes for Annie include +15 ap from quints as well as 7.83 Mpen. These are the exact same for Blitzcrank, so we'll compare with the same rune set.
There's a difference on items though, as Annie's most common starter is spellthief (adding 5 ap and 30 true damage for three procs) contrasting with Blitz's targon.
Let's calculate damage before reduction :
Blitz : 440 + 2*15 = 470 Magic damage
Annie : 395 + 2.4 * (15+5) = 443 Magic damage.
Once you add 3*10=30 True damage from spellthief, Annie's Q-W-R burst is greater than blitzcrank's Q-R.
This somewhat debunks your claim that "Blitz Q-R combo at 6 alone does more damage than Annie's R-W-Q". By something like 15 damage after reduction at most, sure, but facts are fact.
You might want to dispute my claim all THREE spellthief procs are used. I do so because the calculations do not include poke but burst. If we don't include poke, than we suppose no poke was done and we supposed the procs are available for use.
You might dispute the fact I used an item advantage on Annie to justify more damage dealt, and you are right Blitzcrank's base damages are higher, yet LoL isn't played without items, so it's normal to calculate with items bought at that stage of the game (supposing no backs before 6).
Finally, you might say that adding E definitely tips the balance in Blitzcranks favour, but that's where Blitzcrank's damage actually starts to falloff compared to Annie, as he is truly a burst caster while she has sustained dps in Tibber's AoE + auto.
Doing the calculations quickly, one level 6 Blitz E does 75*2 = 150 Physical damage.
We'll suppose Annie gets to do One auto, Tibbers does one Auto, and his fire procs only once.
That's 61 Physical damage + (20 + 0.2*AP) + 80.
Adding brutly before resistances, she's doing 161 mixed damage before AP ratio. Of course calculating resistances favours her because there's less magic resist at those levels and both champions are using Mpen runes.
To conclude my post. You're right Blitz has higher base damage, but he does not do more damage at level 6 than Annie, especially less when you add E.
I'll add though that Blitzcrank's support winrate is at 52.63% (bordering on OP) averaged on Plat+ games. If your statement means that you find Blitzcrank stronger than Annie support, you are right. I myself don't play botlane and have no opinion on the matter, I simply enjoy basic calculations and providing facts. Enjoy the rest of the conversation.