I threw together a cobble generator in a test world. I gave it a pickaxe, then used my own in an attempt to teach it what to do. The golem just makes noises constantly, rotates its head every time I break the block, and occasionally raise its arms in the air. It will not mine cobble.

I also tried to get it to harvest wheat, but it just wanders around even after I used the necessary tool to teach it.

latest.log

I'll mark this as a bug. They should work in 1.21. Pretty sure I did test it in that version. Can't check right this moment though.