"Jack" still implies mediocrity. See also, previous attempts to make a true Jack of all trades Bard.
Two separate issues.
Jack does not imply mediocrity, it implies competence. You seem to think that anything short of specialized mastery is mediocre, and that is simply false. General mechanics are
more useful and
more well regarded than specialists. General Practitioners are equals to specialist physicians. Lumberjacks are professionals in a trade skill.
The idea that “Jack” means “mediocre” rather than “common in the classist sense” is a misunderstanding based on a modern social refusal to acknowledge class structures. A Jack in this sense is more akin to someone who is a professional plumber, electrician, and carpenter, with full training and competence in each field, compared to a specialized engineer. One isn’t “better” than the other, they just do very different things.
To the other, wholly separate issue: The Jack of All Trades Bard has worked quite well several times, it’s just harder to design well than a simpler more specialized class. In 5e, the Rogue is also a decent generalist, because it has cheats for the action economy. What the Bard (or ideally the Ranger IMO) needs is similar action economy workarounds to allow it to do multiple jobs in a turn.
The other problem with some iterations of the Bard is the false idea that a Jack must be mediocre being translated into mechanics in a level based system, putting them at limited progression in several areas. This part of the issue would be
worsened by making the class a half caster.