I put some aero bars on my road bike recently.
I had to start pretty slowly. It was pretty unstable getting onto them, shifting (standard shifters), and things like that. But after a while I got used to it and actually prefer to be down on the bars whenever possible. I probably will get some bar end shifters too, so that I can shift without moving (a convenience thing, not stability).
It takes some getting used to, but I don't find it unstable now.
Maybe your mountain bike does not have a geometry conducive to being down on bars?