Linux terminal ->HDMI->Thunderbolt converter->iMac?

Which vendor and which model server? BMC, iLO, DRAC, IPMI hardware? KVM hardware?

Some Linux servers offer iLO (HPE), DRAC (Dell), or similar remote management. If your Linux server includes that or can be upgraded to include remote management, that’s usually the best and most flexible path. That will offer full remote management, including remote power management. (I routinely administer servers located on other continents using these features, including installations and upgrades and remote troubleshooting.)

If you’re on server gear lacking remote management hardware options, then a network KVM is an option. That’s effectively what you’re envisioning, though yes, you’d probably rather plug in the cables directly rather than through a network connection.

The cheap way to do what you want is a USB-to-serial adapter, and a serial cable into the COM1 console, if your server includes a DB9 console. I use a Plugable adapter for that, though there are other adapters with support. The Prolific PL2303 is fairly commonly supported. This assumes you have a DB9 console port, obviously.

If your Linux distro boots into an ssh console as many can do and that’s really handy for booting in a hosted environment, that’s the cheapest path.

If the only time you use a monitor on the Linux system is to install a new OS, then you could create another minimal bootable Linux installation on that Linux system in a small partition. Then ssh into that minimal system so you can run a Linux installer to use on the main partition. If you are using a Debian based system, then you can use debootstrap to install the OS to the main partition. Of course if the system doesn't reboot properly you may still need to connect a monitor to the Linux system to fix the issue if you cannot reboot back to the minimal Linux installation. It may take a little bit of practice to make this work depending on the hardware used as it can affect your ability to boot to the minimal Linux install again.

While I didn't do this on a headless system I was doing something similar to have multiple boot options on a single drive. I created a minimal Linux install and customized the GRUB2 boot menu so that GRUB2 would always boot to the GRUB2 config for my minimal install and I would manually add the other Linux installs to this menu. Making the main Linux install the default will allow you to reboot the system normally and having a sufficient pause for GRUB2 to allow you time to manually select the second menu option for the minimal install should allow you to remotely manage this system even when the main install doesn't boot. If you remember to customize the GRUB2 boot menu before rebooting you should be able to manage the headless Linux system remotely.

Not sure on the HDMI to Thunderbolt converter, but...

You can connect any of these older iMac models to another Mac from 2019 or earlier for use as an external display. Learn how to identify your iMac model.

• 27-inch iMac models introduced in 2009 and 2010: Use a Mini DisplayPort cable to connect your iMac to a Mini DisplayPort or Thunderbolt or Thunderbolt 2 port on the other Mac.
• All iMac models introduced in 2011, 2012, 2013, and mid 2014: Use a Thunderbolt cable to connect your iMac to a Thunderbolt or Thunderbolt 2 port on the other Mac. If the other Mac uses Thunderbolt 3, you can use the Apple Thunderbolt 3 (USB-C) to Thunderbolt 2 Adapter with your cable.

Target display mode requires macOS High Sierra or earlier on the iMac used as the external display. It does not work with macOS Mojave or later.

FreeNAS might be a good fit, then.