Update: Henry Babcock followed up with me by email on May 6. The Ezoic team had completed an import of the DNS settings from GoDaddy into Ezoic, and Henry reminded me to proceed with integrating the site.
So on May 6, I switched nameservers from GoDaddy's defaults to Ezoic (octopus and tarsier).
This resulted in the site (www.launchexcel.com) going down and my CMS host sending me a warning email to say that the domain was not properly integrated. If this is not fixed in 14 days "the domain will be deactivated and all the connected communities will be deleted."
See attached screenshots for confirmation.
As much as I want to integrate Ezoic, it appears that my current hosting does not technically allow this to happen. This is the second time I've tried, and each time the site has gone down.
Therefore, I have decided to switch the nameservers back to GoDaddy defaults. Now the site is served as normal. This is not a satisfactory state of affairs.