Thanks for sharing. A couple of additional input from my point of view to onboard new team members (new designer)

1. Playbook(s) — Each function should have their own playbook for new comer to read and understand the team better.
2. Shadowing — However, playbook alone is not enough, the next step should be continuing to join important meetings, sprint planning, kick-off, customer video call etc. I find that this is the best way for he/she to quickly understand the overall team dynamics, its users and of course, the product. I suggest to let this person to do shadowing for at least 4 weeks before actually assign them to do real work. Join any calls, meetings, lunches as much as they can.
3. Do Onboarding Retrospective — How was the entire onboarding process like to you? What was good to you that we should continue doing? Okay, typical retrospective scheme here.

How to measure?
It depends on the seniority (experience) of each person. Some able to fully grasp the entire product within weeks while some takes months. It also depends on the complexity of the product as well.

Thanks for sharing again.
Cheers!