> What will go wrong if one stores a Context in a struct?

Contexts are about the dynamic contour, i.e. the dynamic call stack. Storing the current context in a struct and then referring to it in some other dynamic … context … is going to lead to all sorts of pain: timeouts or deadlines which have already expired and/or values which are no longer pertinent.

While there are some limited circumstances in which it may be appropriate, in general it is a very strong code smell. Any code which passes a context should receive a context. And any code which may pass a context in the future should receive one now, to preserve API compatibility. So any exported function really should have a context as its first argument for forwards-compatibility.