By the logic of "keep B- weapons or stronger capped even if the party's better off if you don't", we would get the following:
WAR neglecting A+ GAX and A- Axe to cycle through: B- Dagger, B Sword, B+ GSD, B+ Scythe, B- Polearm, B- Club and B Staff.
MNK neglecting A+ H2H to skillup B Staff.
RDM meleeing, period (with B Dagger and B Sword).
WHM meleeing, period (with B+ Club).
BLM meleeing, period (with B- Staff).
PLD neglecting A+ Sword to cycle through B GSD, A- Club, A- Staff.
DRK neglecting A+ Scythe and A- GSD to cycle through B- Sword, B- Axe, and B+ GAX.
RNG meleeing with B- Dagger or B- Axe, rather than standing back and shooting stuff from a good distance with A- Archery/Marksmanship.
BRD meleeing, period (with B- Dagger).
BST neglecting A- Axe to skillup B- Scythe. (Though if they're solo and they want to, more power to them.)
SAM neglecting A+ GKT to skillup B- Polearm.
DRG neglecting A+ Polearm to skillup B- Staff.
SMN meleeing, period (with B Staff).
BLU neglecting A- Sword to skillup B- Club.
COR meleeing with B+ Dagger or B- Sword, rather than standing back and shooting stuff from a good distance with B Marksmanship, and running around keeping rolls on people.
(PUP does not possess any offensive combat skills B- or higher, unfortunately.)
What can we conclude from this? There are times and places to skillup weapons you're kinda good at, but which are not your best contribution to an exp party. (Whether the rest of the party is pulling their weight or not is irrelevant to this - the solution to that particular problem is not to bring yourself down to their level and consequently lose any right to criticize.)
Hint: think skillup parties and Besieged (with the provision of "only if it's going well").
If your second best weapon caps significantly lower than your best weapon, if it has even worse WS selection than it (and while it's hard to have worse WS selection than Sword, Club manages to do that), and if there are practically no weapons of that type usable by your job that actually have good stats (most good clubs are PLD and/or WHM, only) ... yeah ...
Don't melee with a club in an exp party. You could ignore skilling up the weapon entirely on BLU for all that it'd really affect you.
EDIT: As it is, WARs and DRKs and RNGs have a hard enough time keeping their justifiable multiple weapons (A- or higher, and good weapon and WS selection in each) capped in exp.