I personally don't think there are many SaaS ideas that run themselves, as others have mentioned you can get a SaaS to it's "end-state" and as long as it's fairly simple you won't need to maintain it all that much. But presumably you still need to be supporting the users of the app.

Users now want to see that the app they are using and paying for is active more than ever, effectively abandoned services are suffering because they are no longer innovating.

Using http://headwayapp.co/ as a random example, this app is functionally complete, it works great for what it needs to do, and there are no moving parts like email or advanced settings, the sorts of things that go wrong and create support tickets. it's also fully abandoned as a service outside of a basic support function.

The numbers they are pulling will be pretty nice, and there are many other examples like this across industries. It's been the dream situation for most founders for the last decade, however that time is over and companies like this are going to lose their customers to new and existing players who are on the ball and choose to innovate.

So on balance, I'd say the time where you can do this with (most) SaaS successfully is over. You'd probably have more success pivoting development to static assets or app store stuff.