I found the following issue:
if I set a simple built-in model material for a model having an animation, I get an error:
ERROR:GAMESYS: Unable to bind bone matrix cache texture for component '/builtins_model_local', no free texture slot available.
Well, and also with my model (made in Blockbench then re-saving .glb in Blender) skinning breaks something, maybe some information is missing in the model or it is in the wrong format. Another interesting thing is that if there are two identical models on the scene, one has no animation set and the other has it, both play animation. See the video: