I doubt devs actually say that they can do testing themselves. It sounds like something manager says.
The reason is what you wrote later: it's not fun and they don't do it properly (e.g. full test case instead of testing just changed part) and they know it.
I've been on two test teams and three dev teams... I think devs can do it, and I say as much when I'm in dev mode.
The manager just has to recognize that 80% of the time this will mean a 20% increase in dev time while the tests are written, and 20% of the time it'll mean a 400% increase while things are refactored to remain testable.
It sucks. The rest of the business doesn't get it. But the alternative is that you end up painted into the same corner: in need of that drastic refactor, except the people who it's blocking (QA) don't feel empowered to do it, meanwhile the devs just keep shipping features because they're not the ones that are blocked.
Better to take that occasional slowdown than to have nobody owning it at all.
The reason is what you wrote later: it's not fun and they don't do it properly (e.g. full test case instead of testing just changed part) and they know it.