AppKit restricts access to the AXClassName attribute and several other attributes to applications eligible for com.apple.private.accessibility.inspection . The XPC service that the Accessibility Inspector uses to access availability information has this right. Since this is a personal right, I believe that it can only be added to applications signed with an Apple certificate.
You yourself can verify that the right is key by removing rights from the XPC Accessibility Inspector in this way (first back up the accessibility inspector!):
codesign -f -s - /Applications/Xcode.app/Contents/Applications/Accessibility\ Inspector.app/Contents/Frameworks/AccessibilityAuditDeviceManager.framework/XPCServices/axAuditService.xpc
If you then open the Accessibility Inspector, you will see that most functions work correctly, with the exception of things like the class name, which is private.
XPC-enabled Availability Inspector

XPC Perpetual Availability Inspector

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