I think it’s always a good idea to start with the basic principle that our planet, or sun, our solar system is nothing special. That is, our solar system won’t be doing anything different than the others, and given the configuration of the Milky Way arms, these stars (and their planets) are not getting “flung out”. So we wouldn’t either. Our solar system is too gravitationally bound to not just SagA, but all the mass, including dark matter, holding us together.

By that same logic, we have to assume the Milky Way, which has very old stars itself, isn’t special than most spiral galaxies we see. It’s bigger than average, yes, but within normal. And those distant galaxies we see are not ejecting their stars either. The galactic escape velocity is very high.

In fact, I bet astronomers have already calculated the mass of the Milky Way, it’s escape velocity, and the speed at which our star and solar system move through it. I think we’re safe.