Thank you very much!
I didn’t know this existed. For posterity here is what I did:
apt-get install nullmailer
echo "smtp.ionos.de smtp --auth-login --user=mymail@example.com --pass=mypassword --port=587 --starttls" > /etc/nullmailer/remotes
echo "example.com" > /etc/nullmailer/defaultdomain
chmod 600 /etc/nullmailer/remotes
service nullmailer restart
Tested with this and with the fake check results command:
echo "error" | NULLMAILER_NAME="Testsystem check" mail -s "This is just a test with nullmailer" "mymail@example.com"