[source: life in Ontario, Canada, previously Minnesota]
Once it gets frozen like this it pretty stays that way until spring or at least an extended warm spell. The amount of salt I would need is pretty cost prohibitive. I even tried breaking it up with a pry bar a few times and quickly realized that was an insane task. As you said I'd never spray the hill down with warmish water for fear of making the problem worse.
If your house was built in the last ~25 years it will have frost-proof outdoor faucets (example [1]), and there is no need to turn off the supply in the winter. Many newer homes don't even have a valve for turning off the water supply (except one single valve somewhere near the house water source). As long as they are installed correctly you are fine, but you definitely cannot keep a hose connected or you risk defeating the protection.
I live on a small hill and have a snow making machine so the kids can ski/snowboard in the winter, and we use the outdoor faucets for hours at a time. But when finished, we disconnect the hose and use a special nozzle to connect the hose to an air compressor and blow out all the water remaining in the hose. Once the water stops flowing it freezes very quickly and the hose is useless unless you bring it inside the garage for a day - and then it makes a big mess in the garage!
[1] https://www.amazon.com/dp/B0051966QW?tag=wnla-org-20&linkCod...
Salt is corrosive however so if you keep you car or bike there you will get increased rust problems if you skimp on cleaning them.