In the early days of the 777-300ER Boeing “Toyed” with the idea of fitting a big APU as a sort of APU/boost engine to improve it’s take off performance. They forgot about it rather rapidly when GE produced the BIG GE90.
On the subject of Boeing the 777’s tail cone is sort of fishtail shaped and the APU exhaust exits side ways but the 797 tail is the traditional shape. Why go back to that shape???

wysiwyg.
what I mean is that anything that pushes air backward produces a opposite reaction how ever small. As you rightly say the APU does nothing for thrust production but I wouldn’t like to put my hand into the exhaust stream!:)

Rgds Cking