This is especially inconvenient because the scene I'm using has an orthographic camera.
Am I missing something?
Play mode uses always the default (perspective) camera of the scene. You can preview it with different cameras (see screenshot) but this setting is not used when activating Play.

You can change the Player's camera by calling setCamera(camera).
player.setCamera(scene.getObjectByName("OrthographicCamera"));
Add a script to your scene that will call this function.
Most helpful comment
You can change the
Player'scamera by callingsetCamera(camera).Example
Add a script to your scene that will call this function.