There is no requirement to upgrade the OS. You can run the system on the older OS, Apple won’t remotely kill the system if you didn’t upgrade.
Now with that said, I always recommend upgrading as best as you can. You may find older Apps won’t work properly or Apple may no longer offer OS support for your system which might stop you from moving forward.
So let’s see if we can diagnose the system… As you are getting the onboard diagnostic LEDs responding we can deduce the issue is deeper in the logic. At this point I would do a careful inspection of the logic board looking for damaged components and if you have a infrared camera see if you have an area getting a bit too hot that shouldn’t be. You likely will need to take the logic board out to inspect the back side as well.
You’ll likely need the schematics and board views for the logic board so you can run deeper testing checking for shorts and checking the power rails.