NO! The bigger the magnet ... stronger magnetic flux to control the voice coil's movement in the gap. Altec's and JBL's of yore were noted for this aspect!
For drivers, no, but in speaker design you can often trade sensitivity for deeper bass. Less sensitive = deeper bass.
A great example of this is the Magico S1 Mk II, which is pretty inefficient (86 dB) but achieves remarkable bass for the size. (sits back and waits for the Magico shills to go absolutely bonkers).
Not many things in audio depend on just one other thing. Being connected, most things depend on being connected to something. But there must be at least a half a dozen things I have done that have made the bass tighter and more articulate but had nothing to do with efficiency. So the answer is no.
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