I'm not familiar with any of the other proposed solutions - could you link to a few?

One important factor is that you need to install the Python Launcher to get the context menus (this is the default in 3.5 and later). Associating directly to the Python installation just causes conflicts when you install another version, so we set them up via the launcher instead. In effect, the launcher is the integration point between Windows Explorer and Python, and without it you only get the shortcuts.

Installing Python 2.7 and leaving the "Register Extensions" option selected will overwrite the context menu from Python 3.5 or later. So if you've installed 2.7 after 3.5 then I would strongly recommend running a Repair of 3.5.

Otherwise, answers to Terry's questions would also be helpful. It isn't entirely clear how you are "choos[ing] the IDLE application", or if the installations were successful.