In truth, they'll be orbiting the galactic centre in an orbit somewhat offset from the Solar System. The Milky Way will have completed about 20 rotations. No idea what the resulting offset would be, but all but certainly somewhere within the galactic disk itself.
Space is big. You just won't believe how vastly, hugely, mind-bogglingly big it is.
I liked Neil deGrasse Tyson's description of how what we call space is still ridiculously close to our planet and how far away everything else actually is, using a standard classroom globe for scale: https://youtu.be/Tt0uV5d8tss?t=99
I wonder how long the gold-plated records will last.
Tyranny of the rocket equation strikes again.
That can be solved with a much larger ion drive, which will be much cheaper to launch (along with appropriate amounts of propellant) using Starship (or its future successors). And if a wealthy individual that has everything wants to retrieve that golden record for his personal collection, and is willing to spend a couple billion on it, well I'm sure that Musk's great grandkid will be more than willing to oblige.
https://www.wolframalpha.com/input/?i=2*sqrt%28122AU+%2F+%28...
Such a device could cross the distance in a few weeks.
Now we only have to go to project rho and build one fitting rocket design ;).