First of all, thank you to all you commenters who shared their method of storing thread!
Seems that I am not uncommon in my approach to re-wind only some of the threads, and on different things.

One fact that I have found is that some threads are better suited to being wound on a simple stick or paper roll or similar item with no flanges at the ends, while other threads still have a lot of "spring" in them (usually because they were overtwisted slightly during the manufacturing process) and tend to spring off the smooth thread holders. Flat thread winders or bobbins/spools with flanges are definitely the better choice for such pieces.

If speculation is allowed, I can imagine thread winders made from parchment, too (and I am planning to make some to try): parchment is light-weight, smaller bits of it will accumulate anyways when cutting parchment sheets for book pages out of the whole parchment skin, and it should be very well suited for making thread-winders. Small bones might also have been used, with the thicker joint ends making for a more secure storage of threads even of the "springier" sort. Storing threads on the spindle (especially just until the second batch for plying has been spun) also makes a lot of sense to me.

So maybe lathe-turned bobbins are that rarely found because there really weren't many of them in use - people preferring to use stuff that was handy, or less hard to get and less expensive. Scraps of parchment, slivers of wood, short bits of smooth twigs, small bones, bits of reed, spindle sticks, or simply little balls of thread with or without something as a core. I know that if I need to wind some thread on something today, I'll just use whatever useable thing I can grab at once, so I've torn off some paper from chocolate wrappers, folded that and used it as a winding core more than once. Seen this way, it sort of makes sense to me to just use cheap, available things that do not scream "thread storage item" to the archaeologist and save the money available for textile tools to invest in things not found or substituted as easily. And as far as I see, the habit of using precious, shiny and expensive matching sets of needlework items only comes up much later - so representation of personal affluence by having precious thread storage items was probably not done in the middle ages.

Hm. Maybe I should have some nice roasted chicken soon and keep those practical little bones...