It's spontaneous because it is not caused by the external treatment. It's not a misnomer so much as a difference in usage. When you say a person did something spontaneously, it means it was impulsive, not caused by anything, and has connotations of being a sudden or easy action. When you say an event was spontaneous, there is no reference to meaning or intensity of the event - it simply means that there was no external cause.

Still, if my physiotherapist tells me to strengthen my core muscles with daily bodyweight exercises and I instead strengthen my core muscles with weekly free weights exercises, if I am cured it would seem odd to call it spontaneous.

The statistical question is whether you would have done those exercises regardless of whether you consulted a physiotherapist. If you would have done those free weights exercises anyway, then it's unreasonable to credit the physio with your recovery; if you wouldn't have done them of your own accord, then it's reasonable to credit the physio even though the precise causal mechanism wasn't what we would expect. We're not trying to address philosophical questions, but find out whether particular treatments have a meaningful effect.

does not vanish?