Are there any alternatives to Raspberry Pi that have integrated power
AFAIK no. There are certainly add ons which can remove power from the Pi.
Speaking as an engineer, I have to ask "What is the objective"?
If it is to save power first calculate the cost of running the Pi. I did this some time ago, and it amounts to a few dollars a year - about what most of your appliances use on standby. The Pi uses little more than a PC which is "off". If you really want to save power you need to control the mains input to the power brick. It would take years to "save" the cost of the external control device.
Is it to protect the Pi? This is valid and running a shutdown is a good idea if you are not going to use the Pi for some time. This can be done from a script and it is easy to control this from a button if desired.
If you are running the Pi from batteries, particularly in a remote location, power control can be important but you need a more sophisticated solution, which should be integrated with a watchdog timer to ensure reliability.