It's not a management issue, it's deeper than that.
Whenever you change something, you risk breaking something. It doesn't matter if you have unit tests, integrations tests, code review, staged rollouts or whatever else. Every time you change something, you risk breaking something. It does not matter how good your QA process is.
> improve processes and legacy code
Refactors are also a major source of bugs, so you should do that only if there is a serious problem with the old code.
The problem is that the code is constantly changing. With SaaS, customers have no choice but to always use the latest version, so eventually they'll run into a bug. Because constant change means that errors slip through eventually.
With traditional desktop software, customers aren't forced to update. They can just keep using their version of the software, and rely on the fact that it's not going to break suddenly.