People say this, but I think there are so many cases (notably coin tokens, durations, and re-shuffle controlling) where Possession really does hurt that it is actually, on average, noticeably hurtful. That doesn't mean it should be an attack card, but I don't think it's fair to say it does no harm on average; that's only the case if your possessor does nothing to take advantage of the fact that it's your deck he's controlling, but any smart player will make sure he hurts you when given the option. Plus, Possession really changes the way you play in most cases; certainly this wouldn't be the case if Possession were just a pseudo-minion attack.
Possession changes the way you play. Yes, you have to take it into account when it's on the board, but that's not a property of attack cards alone. I'll play differently if Embargo is on the board, or a trasher, or alt VP. Most cards, really.
And if you take it into account, you'll be more careful about coin tokens, durations, reshuffles. Yeah the Possession player is going to try to hurt you, but Possession alone doesn't enable that. It depends on your deck composition, which is something you control a lot more than your opponent does.
But you seemed to be arguing that Possession is roughly just a gainer combined with a Minion attack that draws five instead of four (which is obviously not hurtful), and I'm saying, no it forces you to play in what would, in the absence of Possession, be sub-optimal, a significant majority of the time. Either you eat my coin tokens, or I have to use up my coin tokens earlier than I would like, or I have to skip that coin token card I wanted altogether; all three of those cases are things that hurt me. The same way that the presence of Noble Brigand might make me play an engine because I can't let you steal all those Golds, the "attack" aspect of Possession forces me to re-consider how to play a board. Of course what my opponent does might affect how I play for non-attack cards too, but these are almost always in a positive way. If my opponent gets lots of Council Rooms maybe I can focus on getting other stuff instead of draw. How often does the cycling you get from Possession change the way you build your deck positively?
Embargo obviously can't be an attack, and again I'm not saying Possession should be an attack either. And the other stuff you mentioned, you play differently because
you are planning on getting those, not because your opponent gets them. (You still do play differently depending on what your opponent does, the piles situation, etc., but that's more of a property of general strategy and not specific cards.)
But actually Embargo is a really good comparison. Embargo hurts you in roughly the same way that Possession hurts you: either it hurts you by forcing you to play differently than you would like (differently from what would be optimal if Embargo/Possession were not present), or it hurts you by giving you a curse/stealing coin tokens/durations, etc. Either way slows you down, and either way hurts; you can decide which one hurts less and play accordingly, but neither is desirable.
My point is, in a particular game, if Possession doesn't hurt you on average, your opponent is probably either using it poorly, is getting unlucky, or should not have gone for it in the first place. Sure there are a good number of games where it is worth going for Possession and none of those things are true, but my point is that Possession is hurtful on average, because in most games it's bad to be Possessed, and in some games it's neutral, and then in a very few games it's good (only considering games where at least one player goes for Possession, since I don't think games where it's there but both players ignore it are relevant).