Yesterday I finished printing the broken pieces. I made the core out of semi-flex (98 hardness) filament, as that would never break. I also used it, as an experiment, for one of the center pieces (it's cap is still PETG). I still do not have springs but with the slightly flex core there might be enough play for it to work well.
As you can see, the top is in three colors, slightly spiraling counterclockwise going down (with the same pieces it could be assembled spiraling in the opposite direction). The tip and cap are white.
I decided to first try it as a puzzle before risking it as a top. Even though it's topologically equivalent to a Rubik's cube (except for 3 centers not being rotational symmetrical and two corners being so), I find it much more difficult. It's very confusing and I still have to wrap my head around it. I cannot yet follow mentally the pieces and even completing a "face" I find challenging. But it's an appealing challenge, so it's a great puzzle.
On the left there is the solved configuration and on the right two views of one achieved with a few moves:
As you can see 3 pieces ("centers") on the top switched places, moving counterclockwise (the same happened with the 3 centers on the bottom.) The yellow one remained oriented the same way with respect to the top axle, but the other two spun around and because of their weird shapes the top now has two horns, even if all the pieces are in the correct positions (not counting colors).
With a few more moves, I got this animal-looking configuration:
I found myself giving different configurations different names based on their shape.