“Is it necessary for a Stoic to be constantly striving toward productive activity?
Like a religion, Stoicism is paralyzing for its practitioners. This is akin to needing to perform daily prayers to hit the ideal performance level.
I've described (contemporary) Stoicism, and more accurately its current standing in regards to its adherents as a "a cult of perpetual self-improvement". Many in that crowd see nothing wrong with that statement despite picking up my pejorative intent. In my eyes, self-improvement or whatever sort of sharpening of your abilities that also encompasses must serve some end, or at the very least be aimed at a general purpose rather than itself for the sake of it.