???

Grail is an extremely OP item compared to AA (except in cases I stated in my original post).
Not sure how buying an near-incomparably efficient item over another is unconventional.

For Chalice to take effect, you have to be low on mana.

Yes, after you run low on mana, it generates more than TotG. However, mana costs and mana pools on champions are wildly different values. You may have one who has a large mana pool and low costs, or another with a small mana pool and insane costs (Sion).

Sure.
But chalice starts with 7.5 mp5 and tear starts with 7 mp5.
...and chalice unique passive applies to all of your mp5, which means the mp5 would beat out AA's mp5 earlier as you drain mana.

Like I said in my original post, there are champions with high costs and a low pool. Champs like these will need BOTH AA and GRAIL. (Kog's ult can cost up to 400 mana per cast with a .85 second CD)

sion roughly have 7.5 mana regen at level 3. Whit Chalice it's 15. At 0 mana its 30 per 5, so 6 per second. In 58 seconds at 0 mana you get 350 mana tear has and well, in the same time tear grants you 87 mana not taking into account base mana regen, so Chalice need 14.5 extra seconds at 0 mana to grant equal mana than tear. So in the first 72 seconds, either spamming or not, either whit 0 mana or not, Tear is better sustain and not taking into account the Mana from the passive. Also, having large mana pool means you can chain better you'r skills. Or you want to tell me that AP Trist (or even AD but she don't build mana) benefit more from having more PASSIVE mana regen, than having the possibility to chain her skills at least twice?

Want to talk? Use numbers. Math. Don't talk bul**** whitout proof, because my numbers say that Tear is better at least during the first minutes of the match. And late game, it's still better on low pool champions, like Sion, Trist, Miss Fortune, Karthus, Kog Maw, Anivia... etc etc.

You would run mp5 yellows on most caster rune pages (+ ~3 mp5) with a 9-point down the utility tree (+3 mp5). On PG, you start at lv3 so sion will have a larger base mp5. Try your numbers again.
I already said its good on certain low mana pool champions.

It's a large mana pool vs +12% mana returned on kill/assist. I don't understand the "chaining" term.

Chaining is when you spam you'r abilities togheter.
And I stated sion at level 3. Also, not everyone has mp5 yellows.

Well, in Proving Grounds, you'll have to die in order to buy anything, so some people might prefer good burst potential over sustain...

On AD champs who need a lot of mana, though, I wonder if Tear would be worth it... there aren't that many options for items that give AD and mana (of course, that doesn't matter for AP casters).

And it interacts with the Strength of Spirit mastery, if someone is using a wierd build that takes advantage of that. But that seems unlikely ('cept for Ryze, of course).

Overall... Athene's seems to be best in most cases.

It's Proving Grounds. In other words, who cares what they build?

I build for late game. **** ahri doesnt even pass 100 AP most of the time until lvl 12

Taking this logic of sustain being better no matter what, anyone not buildling triple regrowth is being bad.

Because you can't kill someone without committing all in, and if you don't, constant-health-potion-of-regen means you lose as soon as your pots run out.