If it fails it’s healthcheck, then it reports unhealthy. Check and see what it’s healthcheck is configure to be.

Doesn’t look like anything integrates with it yet except for something called Mastopod.

Can’t tell if this is for real.

I really think you have conflicting resolvers running on startup, which would explain this. Double check your systemd units that are enabled on boot. If you don’t see anything like networkmanager, reboot the machine, get the status of systemd-resolv to make sure it’s actually running after a fresh boot, check the logs and see if you see anything interesting there, then restart it and check the logs again once DNS works. Something is different between those two actions.

See if this helps at all:

sudo systemctl revert systemd-resolved
sudo systemctl restart systemd-resolved

Also, what does ls -lh /etc/resolv* show?

What’s happening in journalctl -u systemd-resolved?

Well only your DNS is broken, so that’s all that needs to get fixed. Are you POSITIVE you’re using systemd resolve and not networkmanager?

Did you insert the sysctl values and reboot?

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Synology sells single bay units now, but anything that has network connectivity should work fine. If this is just off-site backup, find whatever is the cheapest. If an RPi with a USB attached disk is cheaper, do that.

You can find MiniPCs of all sorts really cheap now, though almost all have smaller onboard SSD storage.

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Doesn’t seem so.

And there’s a difference between utilitarian and idiotic as well. The fact you can’t tell the difference is a “you” problem, friend.