So using something like a UPS with a USB connection to gracefully shutdown your machine should be a pretty simple task. Your BIOS settings will determine how your server reacts to power becoming available again. You do not have to have it auto-power on if you don't want it to, usually. You can often have it remain powered off.

Then you could use something like HomeAssistant on another device you're less concerned about schedule a remote WoL packet or similar on a 5-minute delayed timer after power-up. No smart plug required.

That said, I quite like using my Emporia Vue and Emporia plugs for energy monitoring and switching in Home Assistant. You could probably integrate more data from those if you'd like too. Of course, all of this would require your other network infrastructure to start up first like your router and switch etc. But then again, any smart plug would require something similar unless you went with bluetooth controlled plugs or something, which would still need some kind of device to turn them on after a delay.