First off, why does it SEEM like propane boils off first ONLY at cold temps? Because at >50F, you don't miss the propane. More of its mole fraction (and weight fraction) boil off regardless of ambient temperature, but you don't have a problem maintaining a vapor pressure of the remaining mix of fuel if the liquid fuel stays above about 30F.
Therefore, if your mental model is of discrete boiling points (all propane boils and then butane starts or no butane boiling below 0C), that model seems to be confirmed by lousy performance below 0C and okay performance above 10C.
Now, onto a parallel example that hopefully you have experience with: flambe'ing brandied pears or plantains in a fry pan. You've got a hot pan. You add some brandy. Enough alcohol vaporizes that you can light the vapors on fire and bring a flaming pan to the table and impress your dinner guests. It isn't ONLY ethanol boiling off, but water was too. You can see this in how little liquid remains (brandy is only 40% alcohol, most of it is water). And, even though a lot of alcohol and water have boiled off, some of each still remain. You can still taste alcohol in the dessert, despite what we say to rationalize serving it to children.
A more controlled, instrumented, parallel example: My uncle owned a distillery, making European-style brandies from fruit wines (plum, peach, apricot and, once, an ill-advised experiment with kiwi fruit). The wine is put into a "distillation column" in ChemE speak also called a "retort" by chemists and a "reflux column" by winemakers. As heat is applied, an ever changing mix of water vapor and alcohol vapors (with various flavor elements and oddball aldehydes and ketones) boils off. At first, the vapors coming off are relatively high in alcohol because (1) alcohol is more volatile and (2) there is a decent mole fraction of alcohol in the liquid (10-15%). As the distillation progresses, the temperature of the boiling liquid increases as the mole fraction of alcohol decreases. Just by looking at a thermometer, the distiller can know precisely what "proof" is coming off at the moment.
There's water coming off at all times – you never get 100% alcohol. There's alcohol coming off at all times, too, but in ever-decreasing amounts. He'd cut off the distillation while there was still more alcohol in the vapors than in the original wine because he wanted to avoid some higher-boiling, bitter flavor compounds, but that doesn't effect us with our stoves.
The bit to focus on in this example is that as a larger fraction of the more volatile component (alcohol/propane) boils off, less remains in the liquid phase. When the liquid phase contains less of the volatile component, the temperature needed to boil the liquid mixture rises. This is bad in cold weather operation.