The charger seems to properly sense a low battery condition, as evidenced by the fact that it turns on when battery voltage is low. But I'm not sure what happens if the battery is disconnected, so there is no voltage for it to sense. In more recent systems, the charger continues to produce its nominal 12 VDC, well-regulated, and the 12-volt loads in the camper continue to function normally from this source. But earlier chargers were not so well-regulated, using the battery itself to stabilize the output voltage. If there is no battery present, older chargers produced a higher voltage. In this event, the charger's own internal over-voltage protection might switch the charger off. This would be considered proper, given the charger design, but annoying.
And they worked the same way with over-current. The charger would not attempt to charge a severely low battery since a very high over-current would result. A very high current could damage either the charger or the battery, so the charger would shut itself down.
As a quick question, do you have a way to disconnect the TM AGM battery, and temporarily swap in a known-good car battery? Jumper cables would be fine, and AGM or conventional will be OK for a quick test. Another (easier) way to do this test is to plug your TM into your tow vehicle, start the tow vehicle engine, and see if the TM's inside lights work.
The TM's AGM battery may have a manufacturing or sales date on it, either stamped into it, or on a paper tag. Do you find one?