Having not played with it myself, I'm not seeing why it would be swingy or luck-based. It sounds a lot like Horn of Plenty, which doesn't feel swingy or luck-based. It rewards building a deck that can get a lot of different cards into play at once. Playing one is kind of like playing 3 Horn of Plenties... which is great, but not swingy.
And Tournament is seen as Swingy because it strongly rewards the first player to pull it off. Magic Lamp is just as good if you pull it off second; especially if you do so before your opponent shuffles.
So what am I missing?
Magic Lamp starts in your deck and rewards building your deck more or less the way you would build it anyway, which is trashing your starting cards and cycling a lot, except that normally you would buy more copies of the same cards to make your deck more consistent, but with Magic Lamp, you have to make your deck less consistent by buying uniques instead. There's a bit of a choice between prioritizing a consistent deck and prioritizing the uniques, but it's a crapshoot either way because in the former case, you can either connect your duplicates or not, and in the latter case, you can either draw enough of your cards or not.
It is absolutely crucial to get it as early as possible because the more turns you spend buying $3-5 cards while your opponent has a fully functioning engine (which they have after connecting Magic Lamp), the more they're ahead by the time you connect it.
Horn of Plenty is different because you don't generally buy it until you're already able to hit 5 uniques every time. At that point, it isn't swingy anymore. Magic Lamp is, because it's possible to get 6 uniques before you're getting them consistently — in fact, it almost always happens and then the other player doesn't get 6 uniques because they aren't doing so consistently either.
Tournament is also different because you have to balance getting the Provinces vs. building the engine, and the Prizes are rarely strong enough to decide the game on their own. Someone who just tries to get Princess as fast as possible can easily lose against someone who builds the engine first — the latter player doesn't have the Princess, but they have the engine, which is usually better than having the Princess but not the engine. With Magic Lamp, the way to connect it is to build the engine, and as soon as it connects, you automatically have the engine, so it's impossible to compete with that by playing better.
It's more comparable to Urchin and Sauna than anything else. Those cards still have more of a skill element to them than Magic Lamp, and their effects are nowhere near as strong as getting three Wishes while getting rid of the Magic Lamp, but they can sometimes have the same effect where both players have to open Urchin/Urchin or Sauna/Silver and then one of them has a significant advantage just because theirs connected. The main reason why Magic Lamp is dumber is that the advantage is so much bigger, plus it's a one-shot effect; you can still have a hard time connecting Urchins or Saunas again after getting lucky one time, which can allow the opponent to catch up, but if you connect Magic Lamp a shuffle before your opponent, you just win.