Why isn't my state-trigger-based binary sensor updating when HA restarts?

I am trying to understand why something doesn’t work, and this is the simplest version I can make which demonstrates my misunderstanding.

Why isn’t the following binary sensor triggering when HA restarts?

template:
  - trigger:
      - platform: homeassistant
        event: start
    sensor:
      - name: "my sensor"
        unique_id: my_sensor
        state: "Tomorrow"

  - trigger:
      - platform: state
        entity_id:
          - sensor.my_sensor
    binary_sensor:
      - name: "my binary sensor"
        unique_id: my_binary_sensor
        state: "{{ is_state('sensor.my_sensor', 'Tomorrow')}}"

Why do I expect it to trigger?

My template sensor is set to update when HA restarts. My template binary sensor is set to update when the state of the first sensor changes. According to the documentation, since I left of any “to” or “from” options on the state trigger, the trigger should “fire when the state of any of given entities changes.”

It was my understanding that there is a “last_changed” attribute for sensors (although I couldn’t find much about this in any of the documentation, only from the forums, e.g. here).

So, I would expect that even though the state value (“Tomorrow”) isn’t changing when HA restarts, I would expect a “last_updated” attribute to change, and thereby fire the binary sensor trigger. But the binary sensor trigger isn’t firing on restart. What am I missing?

How? Sensors don’t all automatically update on a restart, and with a template I should think it would depend entirely on the source of the template. Try it in Developer Tools:

In this example, HA was restarted at 12:45.

Edit: Last changed and last updated are not attributes. Docs are here:

I see. So, is the only thing to do to set

      - platform: homeassistant
        event: start

for the binary sensor as well? Or there is a better way?