Does this need "When ~ ETBS, gain control of all permanents you own"? If I added "If a permanent you own would ETB under an opponent's control, it enters the battlefield under your control instead" as reminder text, would I be telling the truth?

Hmm... No, I think as written, if an opponent casts Bribery or Desertion or whatever, the card stays in your library.

On the other hand, I think teeechnically it doesn't need the ETB trigger, because I think it'll override all "opponents control" abilities. Which seems to contradict my first answer.

Wizards would probably print it with the ETB ability, whether or not that actually is required, as it'd help alleviate confusion.

Well, I agree. Adding that clause will certainly make it less confusing, whether it needs it or not.