Absolutely insulate too! It’s really cheap and easy (if you have exposed pipes in places). Heck, I insulate my cold water in my attic because it’s otherwise a naked run in my attic and that sucks in the summer.
My CA weather privilege is showing (that said, energy and water are crazy expensive here too). You’re making me recall all the hell I went through trying to modernize the 1922 house in MN 15 years ago.
I do wonder how much difference it really makes in energy since we’d be evacuating at least as much water from the tank that it would have to heat fresh anyway. We use the smart plug to set things up for time of use: 7am for showers, 11:30 for lunch, 6pm for dinner, etc. That way we can avoid a constant energy suck. I set things up for a 1-click run for the in between times too.
And shake it all up.