How can I improve the positioning of new windows?
Writing a new window manager or hacking finder is hard to override this very intentional design decision that's baked into the system.

Third party window managers are much easier to implement than to change existing code that Apple intends to own/patch/update. Also, you don’t have to fund (in time, expertise or $) all the development effort yourself as many are window managers are inexpensive or free (even though you expressed the desire to not add a third party solution). 

I really like divvy for the use case I think you are seeking (and liked it as well as Moom in the past)

What Window Management Options exist for OS X?

Also, terminal lets you save window sets, so if you just want one layout to start, that’s something easily done without a third party app.