I agree with all of this but the bass in question only has one pickup.Oh, yes, new pickups can certainly hurt. They may shift the sound in a direction you don't like while costing you and arm and 1/2 a leg.
Work with the controls. Solo the neck pickup. Roll the tone down about 1/2 way. You may find you simply need a new setting to get the tone you want. At the very least you may learn more about what you don't like about your sound and be better informed when you're trying to find different pickups.
The value of the pots can affect the sound. 500k pots instead of 250k will allow more top end to come through. But that's really the only thing in there that makes much of a difference. I cannot hear a difference between a tiny ceramic cap or an orange drop or paper in oil. If it provides resistance then it's all the same.Pots, wires, caps - Of all the things you can change on a bass to alter the sound, I put these dead last. I don't know how wire can be junk. The cap may be junk in that it's actual value is out of spec but unless you turn the tone pot down it won't matter. Junk pots? Perhaps. If they work and are not scratchy, new pots are unlikely to sound better.
The issue I most commonly have with pots isn't tonal, it's that they turn too easily if I brush a finger against them. I like a little more (physical, not electrical) resistance in the turn. That doesn't affect the sound at all though.