Not sure if this will help or correct my running issue but it can't be helping .
Low battery voltage would make it difficult for the fuel shutoff solenoid to work, so you may be on to something. I have no idea how long it would run/mow if the battery wasn’t charging. I wouldn’t think very long, but who knows?
I replaced the VR on mine with an Amazon special. I used the “Holdwell” VR. Looks exactly like the JD part for a fraction of the cost. I used a DTM Deustch connector and also installed the Relay to restore proper voltage sensing. @chuckv has a good pic of how to do the relay