In my opinion, the tone indeed gives hint to the meaning, but I doubt it has a strong correspondence. As to your intuition, namely, the stressed tone corresponds to "know how" and the lighter one corresponds to "will".
Whenever there is an ambiguity, rather to say we figure out the meaning from tones but from the context. Since it's the context that matters. As a listener, we know the speaker add their tones based on the context to convey what he/she means exactly. For me, the context is the root and the tone is just a derived aspect.
Suppose such a scenario. A father wants to let his son finish the homework, however he doubts if the son will really do it. And the mother may come to say, he will finish his work. 他会写完作业的。 So it's natural to stress "会" based on the context, which doesn't necessarily means he knows how to finish his work.
Similar for 要. But notice that 要 is really similar to "would like to", so it can mean a desire "want" but also a tendance to do something in the future (caused by the desire) "gonna"
Edit:
@Sanchuan constrastive stress and neutral declarative context sound new to me. I am not major in linguistics, perhaps I treat the phenomenon naively. For me, these concepts seem to distinguish two kinds of scenario: a neutral one and an unusual one. However I still think the context can explain such difference. In practice, the usage of know how is less frequent than the usage of "will". So the latter may turns out to be a common one and neutral scenario. In contrast, the former contains more information (measured by how unusual the meaning is) therefore needs a stress.