Aren't most SaaS's not pure software, but sell data, communication or offline product/service, and in that way differ from installed software?

The difficult programming in games seems mostly deferred to engines (which are becoming cheaper and nore commoditized), and the game is mostly "content" - more akin to a movie than software. But you're right, games are software, and are thriving.

And games have largely turned into a service as well.

I got two games as a gift recently, and at least one of them require a big download to even function. And both require that i register them with steam before i can play them at all.

Effectively even "offline" games have just become a client to a "service" running in the clouds.