Water – Are the cold and hot water circuits typically isolated in residential plumbing

If you turn off the water supply to a hot water tank, you should also turn off the heat so that there isn't a chance of heating an empty tank. Leaving the pilot on is fine, but leave it set to pilot only.

Other than that, it's a fairly standard practice to shutoff all the water, not just the hot water, in an unoccupied vacation home. I would recommend a 1/4 turn ball valve, instead of the traditional gate valves, because the latter are prone to failure when used. It's also important to set the thermostats in the home to keep it warm enough to prevent any pipes from freezing inside the walls if your location gets that cold.

Remove screen on faucet and take apart and clean. May even have to pull valves on faucets and clean. During your repairs I would guess you disturbed some crud and stopped up the little water conservators in the tap. As a old restorationist I see it all the time.