The built in LEDs on the battery only give you a useful reading when the battery key is in the ON position. For any other key position they give you a reading of a moment of residual voltage and then nothing after that. BTW, I frequently forget to turn the battery key OFF before charging. No problems in a 100+ charges.When the battery is charged and not locked out the lights on battery light up when I push the switch. When the battery is not working or locked out the lights do nothing.
From your various posts above, it kind of sounds like you have a damaged battery or BMS*. If the bike was dropped or banged around too much in shipping that could happen. You've already mentioned removing and remounting the battery, so that rules out loose contact with the rack socket. I'm guessing there's broken or intermittent contacts of some or most cell packs with the main cell bus within the case. It sounds like you're riding on a fraction of the full complement of packs.
The intermittent charging behavior is the biggest tell to me, it suggests there's an internal battery connection that is making contact or releasing depending on heat expansion. The charger getting warm is no big deal; it can normally get warm enough that it is uncomfortable to hold in your hand.
*BMS: Battery Management System, the circuit card within the battery case that manages your cell packs for you, mostly to keep you from damaging them with excessive use scenarios. None of the riding you've described is an excessive use scenario