At this point the web is about as complex as an operating system in terms of complexity. That needs really strong specific standards in order for it to work, and in turn projects like web browsers are huge and complex.
If someone wanted to build a web browser that only followed the simpler parts of the specifications, it wouldn't work for many websites* and people would not use that browser.
*Whether or not sites need to be so complex is another question entirely, but the reality right now is that they are
100% online games in the past were perfectly playable even after developers / publishers ended support. Online only games dying is a relatively recent invention. This petition is asking for consumer protection to return to the norm where a purchaser of an online game always has the choice of being able to play it in some fashion.
A game developer could do this by releasing a server application. They could even do this at the barest minimum by releasing documentation describing how the server ought to work, to allow for reverse engineering.
The Stop Killing Games campaign as a whole isn't asking for perpetual server access, just to ensure that games stay in some sort of playable state.