As John Lawler says in his comment, there is no real difference in the basic meaning of these two phrases that hold in any given context.
There is, however, at least one context where there is such a difference, in that only the version with to would be likely to be used: if the hinges in question are absent or seen as detached from the door, rather than as being an integral part of it. For example, if you’re assembling an IKEA cabinet, you might say:
I can’t find the hinges to this door here, have you seen any extra ones lying around?
Since the door is not yet installed and the hinges not yet in place, they are not really the hinges of the door: that implies some type of genitival, vaguely possessive relationship between the two, almost as if the hinges are part of, or at least ‘owned by’ the door. Here, the door and the hinges are completely separate from each other, and the to is a shorthand way of denoting something like “the hinges that are supposed to go with the door”.
Obviously, this situation is not directly transferable to the ‘crux of/to a proof’ example: a crux is something that is inherently an integral part of something, so a scenario where it is seen as something extraneous and unrelated is simply unfeasible, except perhaps for advanced philosophers (and nobody understands them, regardless of what prepositions they use).