That's why I put 'timed' in there. You can program when they shut off so it just goes back to the way it was before. Like when sleeping or out of the house.
A more fine-grain solution is to get a non-timer (cheaper) version of the pump, and one of those Alexa, Google, or Homekit compatible power switches, then not only can you set the time through a smart home routine, but can override them whenever walking out or coming back.
If using a traditional water heater, you're heating the whole tank all the time. And with a tankless but no pump, you're running gallons of clean water down the drain, waiting for it to get warm. It's all a tradeoff, but this, at least, only heats the water circulating inside your pipes and only during the hours you set.
Many pumps come with built-in timers so you can turn them off when sleeping. You can also connect them to smarthome switches and set a routine to turn them on and off only when needed or via remote apps, wireless switches, or voice control (Alexa, turn pump on.)
We found the cost savings to be non-trivial. Main reason I put one in was because we had a teenager who started the shower running, then went away and got distracted. This solved the problem. And with a smarthome controller, it also reduced costs.
Also, those under-sink instant heaters do exist, but they're only good for a single faucet. They won't work with showers and baths.