Is an ice globe the worst possible way to cool a drink (with ice)? Given an alcohol solution and a mass of ice in whatever shape you like, is shaping it into a sphere the worst possible way to cool your drink without diluting it?
If the ice starts off at a sub-zero temperature, it is able to cool the drink without diluting it by first warming to 0 °C.  After the periphery of the ice hits the melting point, it then waters down the drink in order to absorb more heat.  However, if the conduction isn't perfect within the ice, melting could occur before the whole ice cube hits 0 °C, which seems sub-optimal.
I don't know if because it's not just water that's being cooled the thermo changes, or if there's some other important effect I'm neglecting
More important factors are probably the insulating properties of the container and ice, but all things equal, are all those ice-ball claims categorically wrong?
 A: The latent heat of fusion of water is 334 kJ/kg, whereas the specific heat of ice is only about 2 kJ/kg/K, so unless your ice ball is very cold indeed I doubt the warming of the ice to $0^\circ\text{C}$ plays any meaningful role in cooling the drink. Ice is a good thermal insulator, so the melting is pretty much guaranteed to happen as soon as you add the ice, rather than the ice cooling first and then starting to melt, no matter what shape it is.
There are reasons why you might want the ice to melt more slowly. Perhaps you like your whisky cold but not too cold, and you like it diluted but not to get too diluted if you drink it slowly, so maybe there's something in it.
Of course, if you want it to be cold but not diluted at all then you can just put the bottle in the fridge, or use some cheap re-useable ice cubes. It doesn't quite have the aesthetics of a sphere of ice, though, and whisky does benefit from being slightly diluted. Personally, I prefer it at room temperature, and perhaps diluted with a tiny drop of water.
A: The sphere is actually the best way to cool a drink without diluting it. It was spear-headed by japanese bartending (they know their craft) and it is so effective that Macallan (they make good Scotch) has a ice-ball machine that create spheres of ice that fill the tumbler, and you pour the scotch around it. Edges melt faster. An easy way to convince yourself is to say that the ice cube generally melts into a rounded shape, like an ellipse, if it was a perfect cube it would likely melt into a sphere. If you start as a sphere then it won't be melting into a different shape, instead it will melt evenly and less ice will be lost over the same amount of time. This isn't a great argument, but it is well known that edge effects are important to thermodynamic problems such as this, and it is also well known than high surface areas will transfer heat much more quickly. So, minimizing surface area will get you a sphere. 
Edit: Here is an example of the opposite case.
