According to Script Editor, there are Objective-C APIs that can be used instead of AppleScript or JavaScript. Here's an example, from Terminal:
How do I actually use these? Presumably I'd need headers that defined these enums, classes, and methods. Where are those?
Presumably, given those headers, I could also use Swift via bridging, which would be nice, although not mandatory.
You can't.
Script Editor can only show AppleScript dictionaries.
You find the Objective-C API in the Apple Developer Documentation
AppleScriptObjC does not support Swift syntax.