Possession also is terrible in multiplayer games. The classic example is player A goes Gardens on a bad Gardens board, Player B wins the game by playing Possession/the next best option, and player C is simply screwed. If they build a good deck, Player B just Possesses their deck every turn, often multiple times per turn. If they build a bad deck, Player C can go for a good deck. Yeah other cards, like Smugglers and Contraband can have similar problems, but there are often far more ways around those issues.
The other big problem with Possession not yet mentioned is its space efficiency. It is pretty rare in Dominion for one or even two cards in hand to be sufficient to gain a colony, let alone gaining two or more per turn. Possession can do it easily.
For instance, getting a province from treasure requires that you draw 3 stop cards on a standard board. Their minimum cost is $15. They take three gains to get. Possession is one stop card. It takes a single gain, and yeah $6P is high, it takes exactly one more turn that a gold in a thin/reliable deck. Worse, multiple province gains just require +action (and as pointed out become insanely powerful with Tr variants). Outpost, at least, requires overbuilding your draw to turn up a province equivalent. You can quite easily gain a province with a deck of Prov x4/Possession. And will gain a prov every other turn up until the Prov pile is empty.
All the tradeoffs in building up a deck go out the window if Possession is remotely viable. How do I draw all my treasures and power terminals at the same time? Who cares, gain Possession and trash the rest. How do I deal with the influx of green making combos less likely to hit? Who cares, gain Possession and trash everything else. Balancing draw, actions, gains, and coins for a Possession deck is so idiotically easy. When decisions become this trivial, strategy goes out the window.