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.
Thank you @David Johnson for actually letting us know that the integration also works with Jira Core and apologies for the initial misunderstanding of the issue. As written in my last comment the problem was indeed that the new version of the app was deployed before having granted access from our side.
Interested in learning more about Jira Work Management? Have questions you want to get answered? Join us for our upcoming live demo where we’ll take you through the essentials of Jira Work Manageme...
Connect with like-minded Atlassian users at free events near you!Find an event
Connect with like-minded Atlassian users at free events near you!
Unfortunately there are no Community Events near you at the moment.Host an event
You're one step closer to meeting fellow Atlassian users at your local event. Learn more about Community Events