All officially supported installation methods support third party integrations, which is what HACS is and supplies. Integrations are interfaces to Home Assistant.
Perhaps you were thinking of add-ons?
Add-ons are other software applications in containers managed by the HA supervisor.
If you want to run other software on the same server then your choices are:
to use add-ons (installation method with a supervisor), or
if the applications you want to run are not available as add-ons use a VM for home assistant and another VM for your other applications. You are responsible for managing the OS in this case. Or
use the Container installation method and run your applications in other containers. You are responsible for managing the OS in this case.
Do not be tempted by the “Supervised” installation method. This is a specific method that has a supervisor (like HAOS) but runs on Debian (no derivatives). It has the most restrictive installation requirements and does not allow other applications to be run at the OS level, only add-ons.
As there are no retro game add-ons I know of your options are reduced to:
a VM for home assistant and another VM for your other applications, or
use the Container installation method and run your applications in other containers (I have no idea if this is available for retro games).
And with everything Tom said in mind I’d probably go Proxmox with HAOS in a VM and whatever you want to run for your retro engine in another… Quicknsearch in Google land found three or four hits on doing retro gaming in Proxmox.
While I wouldn’t try to run FarCry on the other VM. It should be able to do a retro emulator ok on an i3. That’s a 4 thread 6th Gen processor. I wouldn’t try to optimize your database in HA at the same time you’re playing Mario but it should handle it. If you can get more ram though… You’ll probably run into memory pressure quickly. I’d love to give all 8 just to HA. Can you upgrade that or are you stuck there?
Your responses have made me think a bit more and I am thinking I will install HA and use an older pc for the gaming piece but I will play with it a bit more before making a decision.