My Apple TV remote does not charge (light on rear does not light up when plugged in). What should I do?

Basically technical design is not good, if works fine you wont feel, once you have problem than its not user friendly at all, apple wants you to get "Apple care" Or buy new, both cases apple makes money. they are not customer friendly.

There is no indicator light. To check battery level:

Open Settings  on Apple TV.

1. Go to Remotes and Devices, then select Remote.

Charge the Siri Remote - Apple Support