Problem with Apple Time Capsule Being Used to Extend a Wireless Network
It sounds like the Time Capsule isn't functioning properly. It may be worth scheduling a free appt at the Apple Genius Bar. They can tell you if there is a recall or if by some chance the repair/replacement is covered under warranty. Perhaps it needs a firmware upgrade? You can check in the Airport Utility on a Mac or iOS device. More info here on Apple's website.

One more thing to consider... Since you are getting decently fast speeds in one location, but not in another, is it possible that there could be something interfering? Oftentimes a refrigerator, microwave, metal paneling in a wall, etc. can disrupt the WiFi signal—although I would expect the connection to be intermittent and fairly slow.

You may consider switching the two Time Capsules to see if the same issues occur with the other one as well. This can at least isolate it to either being a defective Time Capsule, or some other environmental element. It could also be a defect with the Chromecast. Is that possible?

Aren't wireless issues fun?? :) so many possibilities!

Most name brand wireless routers can work as an extender of an existing network. So, no, it definitely doesn't have to be an Apple product. However, make sure all extenders are in "bridge mode" so that they truly do act as an extender and aren't trying to create their own subnets. If you do get another router, it will not be in bridge mode out-of-box. You'll need to configure that using the router manufacturer's documentation.