Thanks for the advice! It does spin and drain at the same time (slowly at first, then faster as the water empties out). I'll look for drain kit replacement videos. Also thanks for the tip about the metal center in the coupling, the part I used does have it.
One more question if you don't mind: while watching the washer spin, I noticed something else: for a few seconds during the spin cycle (the real spinning I think: after it drains and stops for a second), the water inlet turns on. Not for very long mind you, but it doesn't seem normal! Does it sound like a bad contact in the timer ?