The probe (and disc) are infintesmally small in the grand scheme of things, if there's anything that would reveal the position of Earth it's our own signal emissions, which are well ahead of the Voyager probes (the first radio signals are now ~125 light years away. No idea if they can still be detected among background radiation though)

There's a joke about a speeding BMW that crashes into the back of a hay cart. The driver complains that the cart should had a red rag to signal its presence. The carter responds "you didn't see the cart, would have you seen the rag?"

The info in Voyager is just a vanity plate... or a time capsule. Nothing wrong with that anyway. Some time in the future, humans will locate it and put it back in a museum.

To be honest, I thought that was going to end with “you were going so fast, it turned into a blue flag”

It will take tens of thousands of years for Voyagers to reach only the nearest stars, so I don't think that disk is really a problem. Any alien civilisation that could reaches the probes in the near future will already know about us or will find out soon after anyway even without the information on that disk.