One of the reasons given for the greater nucleophilicity of iodine ion compared to fluorine ion is the greater polarizability of the large electron cloud of iodine. The other reason given is that it is less crowded by polar protic solvent molecules. 

However, when the solvent is changed to a polar aprotic solvent the nucleophilicity order is reversed. 

If polarizability had any significant role to play, the order would've not been completely reversed. It seems to me that the only reason iodine ion is more nucleophilic than the fluoride ion is because it is less crowded by the solvent.

Now, this leads me to believe that polarizability is not a very important factor and should only be used when comparing nucleophiles of the same basicity, but of different size (and hence polarizability) in a polar aprotic spent. 

But why then, for uncharged nucleophiles does polarizability dominate over basicity in determining the order of nucleophilicity (regardless of the solvent used)?