Matt:
You have forgotten a major component in your analysis of solar covers: Forced circulation--the colder lower water is pulled in low down and returned just under the cover where it mixes with the warmer water and heat exchange occurs.

A solar cover performs several functions:
1) Insulation--heat rises, whether it is warm air or warm water. That's because the heat forces a given mass of the fluid to expand to a larger volume, therefore making it lighter--convection. That heat will pass into the air warming it, and losing it from the water. So you insulate the water for the same reason you insulate your roof.
2) Conduction. Opaque covers transfer heat by absorbing light, hopefully at a greater rate than water, and the water cools the underside, transferring the heat energy to the water--then the circulation Matt missed has to kick in, or he will be right--it is fairly ineffective.
3) Transmission. The more light a cover can pass the more heat energy the water directly absorbs--greenhouses have used this principle for at least 200 years (George Washington's Mount Vernon has a wonderful greenhouse the First President used for his botanical experiments). True, some light is blocked, but the trapping of the fluid under the greenhouse (in this case our clear solar cover) more than makes up for that loss.

Evaporation is partly dependent on heat energy, but also on the absorbtion capability of the air--cold air doesn't hold much water, that's why when it's really cold--low teens, single digits or sub-zero, your joints don't ache nearly as much as in that nasty wet high 20's low 30's cold. The air is dry. But evaporation is still at work (check your freeze-dried instant coffee or cup'o'noodles). Evaporation due to non-heating--wind-- pulls heat out of the water. It's basic to the laws of thermodynamics and your fridge and A/C wouldn't work if that wasn't true.(there's more to it, but this is still critical).

Notice that even if your hands are warm, rub alcohol on them and they get cool. Shake them and they get a lot cooler. Put them in front of a fan and they get downright uncomfortable. Wind-based evaporation is endothermic so it drops the water temp. The solar cover prevents that.

OK. the bottom line: I'm now of the personal opinion that solar covers are necessary to keep your pool warm and I personally prefer the clear ones to the transparent blue ones or the opaque.