Grammar – Correct Modal Verb Usage in ‘He Will/Must/Might/Could Be at Home till Now’

I think you could simplify it down to this:

1. 'Want' is desire, and is used to express desire of some type. 'Will' has not lost its relationship to 'want,' as 'will' still has the context of "volition" or more simply, "will" (the two are synonyms). It however is not the same context as "do as what you are willing," in which 'willing' means something like "what you'd be comfortable doing, reasonably." It is not a want.
2. 'Shall' is closer to 'will.' It implies a lot of volition, more so than 'will,' which is conditional. It implies duty, just as 'must' does, but if taken in a modern context, 'must' is more informal and applies to a more flexible range of time.

If you need an example of 'will' in the more archaic context, here's one.

There's three verb tenses here, and they're all different in function.

Present continuous for future plans means the decision is made, and there's every reason to believe it will happen. "Alex is going home at 4 o'clock" means Alex has definite plans to go home at 4:00, with no intention of changing them.

"will"-future or "simple future" are guesses about the future without any evidence that it's right, or dependent on a condition. "Alex will go home at 4 o'clock" is just a guess as to what will happen. It's often accompanied by a hedge, like "I think" or "I guess", or by a condition, like, "If I tell her to, Alex will go home at 4 o'clock".

"be going to"-future is the general future form of English. Almost every sentence about the future could use this form, though it's not always the most natural. So, Alex is going to go home at 4 o'clock could mean Alex has a definite plan, or that it's just a random guess, or somewhere in between.