I haven't played DotA's Ability Draft yet, so I'm not entirely sure how it works with animation and all of that, but it sounds like it would be too complicated in League, considering each champion has a set of animation cycles specifically built for their own abilities with no intentional support for other champions' abilities, so such a thing is going to have TONS of clarity issues:
Say, the Champion's model will be determined by the Ultimate you're getting- So we have Ashe's ultimate and model, but we got NASUS' Siphoning Strike (Q). How will the animation of Siphoning Strike, such a core ability with immense power potential on any possible kit, look with Ashe's bow?
How will Aurelion's Sol Comet of Legend (E) interact with champion models like Ashe?
How will Thresh's hook will look on Aurelion Sol's model?
See what I'm saying?
In DotA there's also one resource pool, Mana, while League has a few exceptions- Energy and Fury/Rage for example, combined in a kit with Mana costs. How would that function?
In League there are abilities that depend on other abilities/mechanics (Kalista's Ultimate on her passive and the Black Spear, Azir's E and Q on his W, Yasuo's Ult on any form of Knockup mechanic, Syndra's passive on her basic abilities, etc).
More importantly, how will the range of the champion's basic attack be determined? Will it be based on the model/ultimate's owner as well? That's also something to discuss about.
Moreover, along with a few fun and awesome possibilities for kits, there are probably many frustrating ones, for either the player themselves, the opponents, or both (Too lazy to give examples off the top of my head, but hopefully you get the idea). :P
-_-
So at the end of the day, Riot has two options:
The first would be releasing the mode without many tweaks, which will probably result in a pretty broken mode with many clarity issues and other weird stuff.
OR they could give it a lot of animation bandwidth, fix every possible mechanically broken combination of abilities and so on, which would require a LOT of resources.