It is definetely possible to use the Trusts to get through the dungeons - in fact, it's quite handy to use to learn the ropes for them so to speak without having to deal with other players who may be impatient. I did Mount Gulg with Trusts as PLD myself and I had no trouble whatsoever. Trusts are slower than a typical player party that is reasonably competent though so if you're wanting to speedrun through Trusts probably are not the best for that. But just getting yourself through that dungeon for the main scenario, Trusts are the way to go.

Bare in mind a trial battle is unlocked after completing Mount Gulg which cannot be run with Trusts sadly so you will still need a party for that.