12v battery reading 10.7v not supplying 5v load
If a lead-acid battery shows 10.7V, it is very likely that at least one of its elements is damaged/faulty, and develops a large internal resistance, and the entire battery cannot hold voltage and supply even 200mA of current. The controller cuts it off. I bet that if you try to charge it, the voltage will rapidly increase to >15V, and the controller will think that it is fully charged, and will stop. This (one cell failure) is a typical fault that has developed over time. You need to replace the battery.