Watch movies from Time Capsule on Apple TV I just acquired a 2 TB Apple Time Capsule and copied over all my movies onto the device, using the method described in this video.

My question is this: Can my Apple TV device access videos on the Time Capsule, and play them on my TV?

I searched the web and found this help thread, but when I try to drag and drop files into the Movies folder of iTunes like it says to, nothing happens.
Yes, but you just need a client that is able to play your video files.
I'm not sure which Apple TV generation you use, but if it's 4th gen you can download an app called "Infuse 4" https://firecore.com/infuse

If you use an older generation you can use another solution by the same vendor (I don't have any experience with it) https://firecore.com/atvflash-black.

This guide mentions how to connect to your Time Capsule:
https://support.firecore.com/hc/en-us/articles/215090977-Streaming-From-Other-Devices