It definetly does auto climb. In fact I managed two make it autoclimb twice in a flight; this is normally possible by the way, you just cant plan multiple steps from the get go. So you can set your initial CRZ to 300, configure the step to 340 and once you climbed to 340, you can configure the climb to 360 etc.
Here is a video of me showing the autoclimbs: https://streamable.com/a8h7vz (Video will expire in 1 day unfortunately)