Simple service to check infrastructure status I have a very simple setup here (solo IT guy): we have a on-prem server which runs some services. They should produce e.g. a daily logfile. I'd like to get a notification if this doesn't happen for some reason. I make a script, activated from cron, that checks this and sends me an email if it doesn't happen. However, this obviously doesn't work if e.g. the entire server goes down, my email script has a bug, etc, so I'd like to have the converse: have a daily check and get an email if it did NOT ran. I do have this with www.duplicati-monitoring.com: I get an email even if my on-prem server is down. I'd like to extend this to more infrastructure tests.
How is such a system called? Are there websites (preferrably free / open source) offering this? I looked at eg nagios, but it seems far too big, and it seems that I need to host it myself.
This is conventionally called a watchdog, deadman, or health check.  Bruce's comment is good, including the part about a raspberry pi.   You could also spin up something simple like monit in a container someplace like GCE or AWS.
Other than setting up an external service or server, there are also a bunch of phone apps that will do the pinging and/or http fetch and HTML parsing for you, right from your phone, providing you with a notification and/or audio alert when a condition isn't met. Since you're a one-man shop, using your own phone isn't substantially worse in terms of being a single point of failure, though you'll want to have a migration plan for when and if you do grow.
The thing I would probably do myself is keep the more complex monit configs on-site, summarize those on a simpler monit config from a stable IP at GCE, with a simple config on my (dynamic IP) phone watching the GCE monit.  That lets firewall rules keep things relatively private other than that summary query from the phone, and when you add a new team member they just need to add that phone app with the one query.  If you're comfortable with leaving the firewall rules more open, then you could skip the GCE step and just have your phone hit a summary statistic on your own server directly.
Back in the depths of time I used to use SMTP mail for this sort of thing a lot; these days I'd prefer the phone app for a more timely notification.