I dislike possession and believe it to be less well designed for the following reasons:
1. It takes a long time to play. Yeah, I know cursing games can take more turns, but possession engines (reliably playing possession multiple times per turn either via multiple copies or with TR/KC) already are a lot of clicks/shuffling. Worse, possession rewards you for doing all the things that slow down the game - going for attacks so you opponent has to self sabotage if they play them (of course this leads to other game slowing bits like using reaction cards to offset this resulting in even more decision points during gameplay) - greening early and slowly dragging your deck to a 3 pile and having to do deck management for two different decks (is it better to play this minion to help my current, possessed hand, or to swap my own hand - perhaps one likely to be possessed in turn). KC/Possession deserves a special mention as being unbelievably game slowing. Possession just introduces a lot of analysis paralysis in the game and a lot of mechanical time increases that are just about never there otherwise
2. It viciously rewards turn order in 3er or 4er. Yeah, I know other stuff like masq, smugglers, and tribute are all position dependent, but one of the great things about Dominion is that, in general, where players are seat relative to each other has limited impact. Because the utility of possession depends on the other player there is a great reward for being upstream from a middling player (good enough to build a strong engine, bad enough not to self-sabotage to make possession worse). Likewise, there is great utility in being downstream from the player who went for something that can't go possession (e.g. someone going gardens or jack).
3. It introduces high variance. 6P is harder to hit than 8; while we often weigh cards with the assumption that every potion could have been a silver, in game potions increase variability as they make for fewer degenerate hands (e.g. copper x2, gold x2 is degenerate with copper x2, silver x3); because you can't substitute 2 coins, action cash, or any sort of trash for benefit for the potion portion of the cost you end up with fewer combinations that make 6P than 8.
4. It is exceedingly unfriendly to new players. Possession requires that you look at your deck in a way very different from the rest of Dominion. Having walked a dozen players through possession, I have never found a player who gets how to play possession or against it on the first several passes - this as opposed to say all of Hinterlands that could be introduced with reasonable success. Including the cards that are just a bit weird to wrap your head around with possession - like embargo, outpost, ambassador, etc. makes it much harder to just deal out 10 kingdom cards and have players get up to speed.
Now none of these are individually missing in dominion - hag makes the game long, tribute hugely rewards play order, plat can add huge variance, and black market can create weirdness for new players - but possession takes all the things that I tend to dislike and combines them in one.