How do I setup application shortcuts to work without opening the menu manually?
Neat trick, never thought about it!

So I tried this and I could repro your scenario, the signature will not be inserted until the "signature" dropdown is invoked.  I found that this had to be done every time you wanted to invoke the shortcut, which is a clear bummer.

I also found that when composing a message right after Outlook starts,  if you select (from the main menu):

Draft -> Signatures (select nothing just let the menu go once the signature submenu is displayed)

Then the shortcut will work for all the messages you compose thereafter.

Not the best solution, but better than having to click that signature dropdown everytime.