Availability feature is turned off by default. In order to turn it on for the specific sensor/switch you need to set initial state of the unit by calling setAvailability method.

Example:

...
HABinarySensor sensor("input", "door", true, mqtt);

void setup() {
    // ...

    // method must be called before `mqtt.begin(...)`
    sensor.setAvailability(false); // offline

    mqtt.begin(BROKER_ADDR);
}
...

After availability is set for the very first time you can call setAvailability in any time. Each time the method is called, the device will publish MQTT message to the broker.