Get rid of Nashor's Tooth cooldown reduction and make it heavy AS/AP DPS item?

Nashor's tooth is signature item for last ASAP champions left in the game, namely those 4 oddballs:

20% CDR on DPS item sounds like hidden power for me. This means if those champs can itemize for DPS quickly they will also have their CD shorter - and they have some pretty meaningful abilities to shorten CD on - but if they are forced to i.e. rush Zhonya's instead, they will also suffer in more than one area. Normalizing CDR gain in their alternative damage routes could help put those champions in good spot - i.e. if Kayle didn't get 20% CDR randomly, CD on her ult could be made shorter in general, making her able to survive bad matchups without being overtuned in good matchups, same goes for Teemo's shrooms and Diana's Q. Also Meddler stated that they will give some more options for CDR for mages, so it might just make sense to make those champs to rely on CDR items if they actually want CDR.

Nashor's Tooth 100 AP 50% AS Passive: 8 + lvl + 15% AP on-hit magic damage

+ + 750g (don't care about this example too much, I think it would be a little too weak like that, but gold value should stay roughly the same on higher champ levels)

This is just a proposition, God forbid I don't insist on it or complain about current state of the item, it just would make sense to me to push it into ASAP DPS item, especially if more CDR options are actually added, items with too much different stats are often problematic i.e. current

bonus fun: make it build from + and allow bonus damage to crit :) zeal items now all have some kind of synergy with on-hits (magic damage or triple on-hit proc) only, well, on-hitters don't want to pay for crit itself