As another option based on your original question, you could use the File Notification component and an automation to send events to that file. E.g.:
notify:
- platform: file
name: mylog
filename: mylog.txt
timestamp: True
automation:
- alias: Write events of interest to mylog
trigger:
platform: state
entity_id:
- sensor.SENSOR1
- light.LIGHT1
action:
service: notify.mylog
data_template:
message: >
{{ trigger.to_state.name }} changed to {{ trigger.to_state.state }}
You can list as many entity_id’s as you like. And, of course, you can make the message more appropriate for each type of entity/event with additional templating.