Zapier does, in fact, now connect to Jira Core (Cloud) using Zapier's "Jira Software" integration.
At the time that I wrote this question, there was an oAuth issue which was causing the "You don’t have access to this app" error.
Now that the underlying issue has been resolved, I can confirm that I am now pulling data from Jira Core (Cloud) via Zapier into other apps that Zapier integrates with.