If you have a weak jab, or lead hook, because it is with your off hand, your training/power are the issue, not the hand you choose to lead with. There are some advantages with being a right handed southpaw, especially in something like MMA where you are then able to shoot off of your strong leg.
Most of the advantages are that you will be less common a foe for people you line up with, all the strong hand forward stuff only means IMO the people saying it should have trained more with that lead (weak) hand to make it better. I know alot of people who are right handed who have devastating left hooks and very good jabs, but they worked on them alot, rather than accepting the quick gratification of using the dominant hand to throw them.