Zigbee does work with a generic controller on Home Assistant and other platforms, and there are >3100 devices that are compatible with zigbee2mqtt, a Zigbee to MQTT bridge that exists to bypass the need for proprietary Zigbee bridges. No proprietary app or Internet access required either, but it was not easy to set up. Here's a list of supported devices: https://www.zigbee2mqtt.io/supported-devices/