You can do the story with whatever job you prefer!
Being a magic user, Black Mages has to stand still to cast their spells. You will be very restricted in movement so you'll learn where it's safe to stand during boss fights. The trade off to this immobility is massive damage-dealing spells.
Bard is much more mobile than Black Mage. It's mobility lessens at level 52 but even then it's not a turret like the Mage. Bard's damage comes from several sources so you'll be pressing buttons in quicker succession than with Black Mage. Bard also has some songs that supports the party, most notably a song that makes enemies take more damage from spells, so other magic users will love you.

Since you already have Archer at 21, maybe you could try levelling your Thaumaturge up to the same level and see which one you like the most? The most important aspect is that you have fun playing your chosen job!