Testing a customer's database upgrade on my internal test servers, I ran into this problem again. The PWA banner says "This site is read-only at the moment."
I could have saved myself some time by actually reading the first paragraph of Brian Smith's post on the Project Support blog located here: https://blogs.technet.microsoft.com/projectsupport/2016/04/07/project-server-2016-if-your-pwa-site-goes-read-only/
I had successfully upgraded from PS2013 to PS2016 and when I went to PWA, I got the message that “This site is read-only at the moment.”
Basically, there are two things to check when this happens. First make sure you have at least the May 2016 updates installed and then rerun the Enable-ProjectServerLicense -Key <LicenseKey> command. Rerunning the enable-projectserverlicense command made the site read/write for me.