I don't think you're missing anything - you can see the test case name, owner and a few other fields but not the objectives. I sort of get why this is: the objective field is a rich text field and it could be messy on-screen trying to show paragraphs of data, but even so, it would be great if it was truncated to the first 50 characters or so.
My guess is this isn't ideal but it is possible to do what you want using the API
Cloud: "Get test cases" Zephyr Scale for Jira Cloud API (smartbear.com)
Server: /testcase/{testCaseKey} Zephyr Scale Server API (v1) (smartbear.com)
Hope that helps and good luck!
