This specific petition was broadened to involve all software rather than just games which is why it mentions pinging home instead of focusing on multiplayer servers.
The general idea of the campaign as a whole is to force publishers to create software with a specific end-of-life plan that would include one of the few possible options:
relase the server software to allow players host them themselves
patch the game to not require company’s server (even if not all features would be functional)
allow people to create their own servers after official ones are dead (think private MMO servers)
Any of those options would come into effect only when the official support for the game were to end.
How exactly would that increase the risk of creating multiplayer games? Private server hosting was a thing for years and the only reason we’re here now is because publishers decided they should be the only ones allowed to do it.