As a few people have pointed out, tanks and white mages already have their own mini-limit breaks, which somewhat makes up for the fact that they rarely get to use a limit break in most content. I suspect that's not a complete coincidence.

I see a lot of problems with adding more effects to limit breaks, though. Aside from the obvious issues like favoring a particular class for their LB, I feel like it would cause a lot of unnecessary conflict. No tank or healer is going to LB right now unless it's going to save the party, but what if WAR's LB had an offensive buff? Maybe WARs would considering using LB at LB1 or LB2, thus a.) decreasing the limit bar for later in the fight when it might be needed and b.) annoying people playing other jobs who feel their LB would have been better. As it is, I already have MNK friends who refuse to LB because they don't like risking losing their GL stacks, and if that is going to be an issue I can't imagine if the difference actually matters.