So, if you wanted to use one you would need to redo all your math and effectively use a non rotating frame in the middle of all your calculations, or add fictit

So, if you wanted to use one you would need to redo all your math and effectively use a non rotating frame in the middle of all your calculations, or add fictitious forces and flexible constants. That's what the Christoffel symbols already do - it's part of the formalism, there's nothing to add!