So, do they believe that the Gates of Hell prevailed?
Good question.
Answer "yes"...
But they would attempt to qualify that, I believe, by saying that 1700 years or so was only a "temporary" triumph of Satan over Jesus.
//So, do they believe that the Gates of Hell prevailed?//
Yes. Their claim of a Universal Apostasy would have to mean that the gates of Hell prevailed.
Although, I have had LDS add an ‘again’ to this, meaning they believe the gates of Hell will NOT prevail over the Mormon church, which is nonsensical since they claim to be a restoration which means the gates of Hell already prevailed once.