Well, finally I got some time to validate the claim that the Space Shuttle could go to the moon solely based on using the OMS.
I did it by using another simulation which is based on accurate physics and numbers, in cunjunction with a corresponding accurate Space Shuttle addon. The starting situation was: payload bay empty and full OMS propellant available of course. I did increase the apogee every time I passed the perigee, by a 30 seconds OMS burn. I did this down to the last drop of OMS propellant.
And this is what I got: an apogee of 9550 kilometers, or in other words an altitude of about 3600 kilometers above earth surface. Taking into account that the mean distance between earth and moon is 384400 kilometers: with the entire OMS propellant available and an empty payload bay the Space Shuttle does not even travel 1% of the distance to the moon!
Now, the claim states that the payload bay is used for OMS propellant supply to reach the moon. Let's take a look at it: the Space Shuttle can carry a maximum of 24400 kilograms to low earth orbit, which is not even twice the mass of propellant the OMS fuel tanks can hold (13500 kilograms). And we don't take the weight of the additionl OMS fuel tanks into account here, less than ever any capability for additional life support. This way, the Shuttle would not even carry twice as much OMS propellant as usual. And with the usual amount of OMS propellant the shuttle only manages to travel about 1% of the dictance to the moon by raising the apogee with an empty payload bay.
Well, I don't think we need any further calculation or speculation to realise that the OMS has not the capability, not even in case the entire payload bay is used for additional OMS propellant supply, to travel anywhere in vicinity of the moon.
Supplementary
Doing it via direct insertion or by increasing the apogee every orbit gives the same result by the way: an altitude of about 3600 kilometers. Only the required time is different: 13 minutes for direct insertion and 3 days via adjusting the orbit to get the same apogee. The required delta v to reach the moon does not change no matter how you do it. And the OMS simply does not provide that delta v. Not even if the "little" (in that relation) payload capability is used for OMS propellant supply.
Also, it took 3 days to get a "max OMS propellant apogee" which is 1% of the distance we would need to reach the moon. This way it would take about 10 month to reach the Moon. And again, no life support is considered here, which would reduce the available capability of OMS propellant support.
So, no way would the Space Shuttle fly to the moon solely based on using the OMS, or in other words reach an apogee of 390k kilometers solely based on using the OMS. Not even theoretical.
