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.
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