Automation will not trigger

You should consider using one or more additional triggers to make the automation more resilient. At a minimum I would suggest a trigger for when HA restarts, since that is a common event that can cause these issues. Beyond that you could add one or more triggers for other points below your threshold to act as a fail safe.

  - trigger: template
    value_template: |
      {{ states('sensor.esp_tester_sensor_1_boiler_output') | float <
      states('input_number.start_temperature') | float - 3 }}