Same problem with similar year ES300. Worked fine for years, but now shuts itself off frequently. After I push "I Agree" it stays on but for a few seconds. If I'm lucky it'll at least keep the Route Overview on for several minutes. I know it's working well when the Audio plays again, but it's very sensitive and turns itself off frequently.
Did you find out what was wrong/how to fix it? I heard to replace the GPS system is VERY expensive.
My Lexus service center wants me just to use a peripheral GPS, but I do not want to plug it in/out every time I get in the car.