Most of jq's syntax is just a mix of JavaScript and a typical shell's. Every expression has an input and output like a shell command's stdin and stdout. Functions work on arguments and their stdin to produce their stdout.

Here's an explanation of the syntax in the command I posted:

  jq '
    # output the value at "Directory" from input object
    .Directory
    # pipe to map (JavaScript also has map()). The argument of map works in
    # the context of each element in map's input array.
    | map(
      # Produce an object where the property name is a string that
      # interpolates the value of the "name" property of this element.
      # Instead of interpolating, we could have also used this more
      # JavaScript-ish (ES5) syntax:
      #
      #   {(.name + " has children"): ...
      # 
      # The property value is an expression that gets the value of the
      # "children" property and pipes it to the expression `length != 0`.
      # `length` (which JavaScript also has) outputs the length of the
      # piped input, and then we compare that with 0.
      {"\(.name) has children": (.children | length != 0)}
    )
    # map's output is a single record which is an array. We pipe that to
    # .[] to make multiple records, each an element of the array. The
    # syntax here is comprised of 2 parts: `.`, which is the input object,
    # and `[]` which is the subscript syntax without an index.
    | .[]
  '

> IMO the ideal solution is something using pure JavaScript syntax

The greatest advantage of the current syntax is the great balance it has between legibility and terseness. I don't think making it pure JavaScript would be better.

> possibly with a library resembling jQuery for tree traversal

Using jQuery in JavaScript to traverse JavaScript objects? I don't know what to say...

Something like this would do the same, if somebody implemented a tool called jsed:

    jsed '
        Object.fromEntries( 
            $("Directory").map( 
                x => [ 
                    x.name + " has children", 
                    x.children && x.children.length != 0
                ]
            )
        )'  

Interestingly enough there is already a tool called jsed which seems to kind of do this... https://www.npmjs.com/package/jsed

Edit: Note the jQuery part of it is for advanced cases like searching for specific nodes in the tree, then navigating back up to the parent. Basically the cases like "<Work>[-1][children]" from the jtc guide, I would write as: '$("Work").parent().find("children")'