But why would the hardware power down if it is constantly searching for a 5G base station, not find one because of OPs location?

Mobile networks have more management functions than WiFi, for instance.

If you're in a high density area, the mobile network will push the device to small range cells (handover), or tell the device which other cells are available in that area (list of neighbor cells).

However, the smartphone is not required to search for a base station all the time. It receives hints from the network.

From practical experience, it’s very common for people to find their phone (iPhone or other) burning through tons of battery when they go out in the sticks, and this is reliably fixed by turning off cellular data. So either it’s not as easy as you say, or they haven’t bothered solving the problem.

If there's no service at all, that may be a different matter entirely. As ndjskska says, the cell network itself keeps the phone up to date with the nearby topology, so there's no need for it to keep pinging for a service that isn't there.