Onan Genset only runs while jumped

As far as the generator is concerned, This is normal operation.. The problem is... Elsewhere.

Onan generators (usually) make only 120 volt AC, they do not make 12 volt DC, that they get from the battery, The fuel pump is electric (12 volt) and the control computer "Eats" off the 12 volt supply as well, If you remove the 12 volt supply ... it dies.

So, possible suspects includ all 12 volt connections between Generator and battery, both Positive and Negative (Ground).. All fuses and breakers on that path, Batteries (Are they deader than a doornail) and the converter, Auto Transfer switch, and all the circuit breakers on and betweeen the generator and the converter.

My favorite is the breakers on the generator if the batteries are dead or disconnected.

The generator has a low oil shut-down that can be activated if the coach is not level or if the oil is actually low. Make sure that you are not jumpering around the safety feature that prevents the engine from being damaged.
It also shuts down if there is not 120 volt AC output from the generator itself.

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It would help to know what make and model of your rig.
Has the batteries been changed. Could the installer have missed connecting a wire.
Yes, a bad connection on either the positive or negative could be the problem.

Along with the cable checks, ensure that the battery is not dead, I know you said you jumped it but a bad battery will also cause this.

Check the ground wire between the generator and the chassis. Same on the other end at the batteries. A poor ground will still give you volts but will limit the available amps. While you are at it also check the hot side.