It sounds like a loose wire or connector in the back of the radio, or some other similar connection problem.
The thought of replacing that radio with an aftermarket product wouldn't even cross my mind. You would be amazed at how many of the vehicle systems run through the radio. That said, I'm sure somebody has tried it before, and it would be interesting to hear from anybody out there that tried it.
And by the way, you make me feel bad when you say "My grandmother's 08 lucerne". With the V8 Northstar and magnetic shocks, this thing is faster and handles better than any other car I have owned. It's a wolf in sheep's clothing.