As you are not a gamer, I'll try to make it simple.

If a game ask for an online connection, is usually for three reasons:

• multiplayer, or some kind of social interaction
• drm, to make it harder to cheat, or redistribute cracked versions of said game
• telemetry, either to know how players plays their game, or to sell you as an ad target

When the publisher decide to stop the online component, to save a buck, it often mean the game stops working altogether because of the DRM part, as it basically refuses to start without the proper authorization from the now defunct server.

The petition do not ask them to keep running the server indefinitely, but rather to

• make it possible to bypass the DRM always online part to be able to play the single player part, if there is one. In most case, it is a simple change to do, a function to modify in order to always return "true" (game can be played)
• allow the end user to self host the server. It doesn't mean open-sourcing it, just to release the server software and allow to point to another server than the defunct ones

In both case, the code already exist, and the changes required are minimal, so why not do it? It costs barely anything to the devs/publisher, and gives the game a second life, even without official support.

But they don't. Mostly out of greed, to push people to buy the newest, micro-transaction infused game they wish to sell, sometimes even the same game with half the content replaced by micro-transaction (Overwatch 2 being the perfect example).
They don't want an older, maybe better game to overshadow their new shiny cash grab.