I've had my Quest 3 for about 5 months. It is on v81.1034. When charging, with the Elite Strap/Battery connected to the headset, and the charging cable plugged in to the external battery, the LED on the headset stays on but the LED on the external battery only comes on for a few seconds after I plug in the cable. If I press the button on the external battery, the LED comes on for about 5 seconds only. The LED on the headset itself only turns green when BOTH batteries are fully charged (I think). As far as I know, the LEDS have worked like this since day one. Both batteries are charging fine.
Lucky you, you weren't chosen by Meta to experience this bug! However, in the past both LEDs on Quest and the Elite Strap were on during charge, not only the one in headset.
I've had my headset and Elite strap for 5 months and don't recall the Elite battery LED ever staying on for more that 5 seconds.
