Neat idea!
...but just having an Alexa at all is a big privacy risk, IMO. You can put all your IOT things on their own network all you want, but Alexa can't do anything without the AWS Cloud - so it's always gotta phone home with your voice command, and then your IOT device has to wait to hear back from Amazon to do anything. Plus if you have an internet outage or AWS goes down (it does happen!) you're out of luck.
I think I'd prefer a passive infrared motion detector coupled with a temp sensor (so if the pipes are already warm, it won't run the pump again whenever I walk into the bathroom)
It's not so much the pump power (maybe 80 watts, which would be $85-$170 year worth of electricity depending on where you live ), but the all of the heat radiating off your pipes keeping your hot water heater busy.