That's not to say software doesn't have bugs. Only that bugs don't appear suddenly.
Most bugs are caused by two things:
(1) Changes to the code that accidentally break unrelated stuff
(2) Changes to the environment (eg. OS vendor releases a new version which changes how some API behaves)
I think (1) is more common.
So I think that the reason why SaaS have problems all the time is that typically devs are changing stuff all the time and accidentally breaking things left and right, not that software is inherently unreliable. And since users can't keep using an old version that works for them, they'll stumble across every bug eventually.