It doesn't need to be room temperature. All combinations of cold, warm or hot roux with cold, warm or hot milk work.
The problem is that the whole process is somewhat finicky, and cooks need some experience until they reach a good match between the speed of pouring and the speed and area-coverage of whisking, so that the sauce does not lump. Once they have it, it only works for a given combination of temperatures - if they were to change the temperature, their usual way would stop working.
This is why most cooks believe that making roux is temperature dependent. Once they have perfectioned their own process, they start prescribing its parameters in recipes when they pass them along. And that's how requirements like "the roux must be room temperature" get embedded in recipes. But if you prefer to learn doing it at a different temperature combination, you can - the only problem is that, if you already learned one combination, your first sauces at a different one will fail until you have relearned. Also, the colder processes are somewhat more forgiving than the warmer ones, as you have more time to react before lumps solidify.
The roux itself is typically smooth, I am not sure how you would end up with grainy roux, especially for the lighter kinds. As I don't remember using grainy roux, I cannot say with certainty it won't work - if yours turns out grainy, try it and see what happens.