Even assumed that your statement was correct.
Wait, what? You still don't get this?
So here you go, relativistic.
The light is emitted inside the ring. The ring and emitter move at radial speed v.
Let's also account for the medium, so light (or the "signal") travels at some speed s <= c.
The signal speed in the lab frame for signal taking less time is u-.
So u- = (s-v)/(1-sv/c^2). If we had s=c, then u- = c.
We still have u-t- = L - vt- , so t- = L/(u- + v)
Thus t- = L/ [ s (1-v^2/c^2) / (1-sv/c^2) ] = L (c^2 - sv) / [s (c^2 - v^2)]
Similarly, u+ = (s+v)/(1+sv/c^2), and t+ = L/(u+ - v)
So t+ = L/(u+ - v) = L (c^2 + sv) / [s (c^2 - v^2)]
And therefore t+ - t- = L (2sv) / [s(c^2-v^2)] = 2 L v / (c^2- v^2)
So with L=2 pi R, you get the same formula.
Isn't it interesting that the signal speed s drops out entirely?
Perhaps I should make youtube videos.