atomic physics - Bohr's model and positronium

I was trying to get an order of magnitude estimate for the radii and energies of positronium using the Bohr's model. I did find a few places where they have used the reduced mass to replace the electron mass in the equations. See this for example. However, I don't see why using the reduced mass works. What is the reasoning behind this?

I tried a different approach. If the positron and electron are in a orbit of radius $r$ about their common centre of mass then the force of attraction felt by each is

$$F=\frac{e^2}{4r^2}=\frac{e(e/4)}{r^2}$$

where working in cgs units makes coulomb's constant 1. $e$ is the magnitude of electronic charge.

So, the electron for all intents and purposes "feels" as if there is a positive charge of magnitude $\frac{e}{4}$ around which it is revolving. And therefore, we must replace $Z$ by $e/4$ in all equations for hydrogen-like species.

This gives results that are scaled by a factor of $4$ from the hydrogen atom. Whereas using the reduced mass gives results scaled by a factor of $2$. Why is this approach incorrect?