But credit card companies maintain records of time and place sold, so unless you travel to a foreign country to make your purchase, you're still leaking your location.
They'll learn those cards were purchased from a Walgreens on Market Street in San Francisco three months ago. If the clerk even remembers, the purchaser said something about being dragged to SF for business, and how they really wish they were back in New York with their kids. In other words, less than nothing.
Then your bank still gets the time and place, unless banks are significantly worse in the US than the rest of the world.