Any fixes to Possession that trigger on-gain would end up extra bad with things like Watchtower, because it would allow you to buy Curses and have them stay in your opponent's deck. Of course Possession is already bad in that same way with Masquerade or Ambassador, but that's a bit harder to pull off; and is limited to 1 bad interaction per copy of the card; while a single Watchtower could mean getting a bunch of Curses.
"All cards gained during the turn are gained to your discard pile" or something like that might work. It would change some things, and it's maybe a little unclear on Nomad Camp etc. but there are already rulings for this.
It should be "Any cards you gain on that turn are gained to their discard pile".
I'm not so sure that would work. Although I can't find it in any rulebook, the idea is that Nomad Camp, Den of Sin, Ghost Town, Guardian and Night Watchman have a different default gaining location than all other cards (which have the discard pile). Then Armory, Artificer, Artisan, Transmogrify etc. can change that. So those cards override the default gaining location. For instance, Artisan gains Nomad Camp to your hand.
If the idea is that Possession sets a different default gaining location for
other cards (not itself), first of all that's a new thing and a little weird. (A "default location" should be the default for a given card; it's weird that it can be changed.) Second, we would then have competing default locations (gain Nomad Camp while possessed), which there's no rule for.
It makes a bit more sense to say that Possession is like Armory - it overrides the default location. It would override Nomad Camp. But we then have the problem of competing overrides. Gain a card with Armory while possessed, does it go on your deck or in your opponent's discard pile?