What you are describing there is basically longwave AM radio broadcasting. Large transmitter, small transistor radio receiver.
Low speed data can be and is sent using such transmitters, it is done by phase modulating the carrier. The data can be used to switch electrical appliances on and off.
Right, but that's only one direction. I thought I put it in my comment, but I'm only curious about bidirectional stuff. The best path forward here seems to be WSPR- like stuff, but that requires larger transmitters on the order of tens of meters.
That's the problem, an asymmetry in transceiver size does not really lend itself to bidirectional unless its relatively short distances like cellular phone networks.