No, all devices should work with other vendors. Some vendors don't play as nice, however. The Zigbee light-link spec should allow all lights to work with all light controllers (eg zigged remotes on the wall). Any "true" hub should work with all zigged devices too. Some minor company's hub may not support all products, but thats if they don't comply with the protocol - ZWave theoretically has that problem.
Zigbee is probably more "open" since the protocol has less certification requirements. The only main bifurcation is that "Zigbee" and "Zigbee Light Link" are slightly different and you may end up with a "ZLL" hub that doesn't support non-light devices.
> And if my Internet connection is down, or even the controller is down, the system still functions with graceful degradation only.
Zigbee supports this behavior. You can pair lights/remotes for example with no hub required. Its usually advertised as a graceful upgrade path instead of a degradation path.