Are you reading my mind???? Just yesterday I was pondering something very similar. Our house has about 3' of crawl space and in the Winter the cold water pipes running under the house drop close to freezing. When we turn on the water to rinse something or a quick hand wash it is damned unpleasantly cold. I don't like to run the hot water for 2 minutes just to rinse my hands, so I suffer the cold water at near freezing. I'd like to say "we" suffer it, but my SO, ain't a willing partner in that.
Yesterday I had some kind of epiphany and thought about water pipe heating tape. I use it on the well head pipes to stop them freezing but it occurred that I could use it also for warming the pipes locally. Just to warm the water to a tolerable level for hand washing. I am going to wrap the last 10' of pipes to the taps with tape.
These kind of tapes are available in shorter lengths, but just to give you an idea...
I'd have to modify the thermostat, in fact just bypass it so it is always on. I can unplug it during Summer.