If you want it REALLY easy, you should be able to write a simple bash script that, when called, pings an arbitrary always online website like google, and if the ping returns an error, sends a telegram message to your phone. you could also store the current state in a separate file to allow for "is now down" and "is up again" differentiation, then use the telegram message timestamps to "track" (loosely used term) up- and downtime. To call the script, add it to your crontab and specify your test interval there.
no, telegram is cloud based. you send the message via a CURL POST to the telegram API, which contains your bot token and message. details are in the linked guide.
Kuma obviously is more established and feature-proof, and will work great for most people who want notifications like this. I just wanted to take the prompt for a "simple alternative" overly serious 😎
But their internet is down, so it'll fail to send to telegram. Realistically it needs to be an external system that is tracking when it receives pings from the home network, so it can show periods where the bash script didn't ping for a while.