An update:
After my 7/11 post, the EDD would bounce between July 18-Aug 6 and the standard 8-12 weeks.
On 7/17, I checked the myaccount page around 7am and got the Tesla Vision prompt. The page had my VIN populated. About 1 hour later I got a text saying that my car was being delivered to the service center (Stockton, CA) with a link to schedule the pickup. I could pick a time between 7/22 and 7/25. As soon as I selected my pickup date/time, I got an email saying that I needed to go to the myaccount and pay the amount due (only $2400ish, the rest is financed) before the scheduled pickup date. After that was done, myaccount said that I would receive the final loan documents 24 hours before my pickup date.
So far, so good!