The story I heard was that U2s were being regularly shot down, but the pilots died. This information was suppressed by the government and new volunteers were constantly recruited to fly over the Soviet Union.
Powers put the kibosh on the whole scheme by being shot down, but surviving.
I’m not sure I’d believe that one.
The basic mission profile of a U-2 was “Takeoff ... climb until engine flames out (around 60000 or 70000 ft) ... glide until reaching engine relight altitude ... relight engine ... repeat as required”.
It was only when the airplane was down near the relight altitude that it was even possible (at the time) for the Soviets to shoot it down.
And that was what Oswald supposedly told the Soviets.
I have no idea what the actual veracity of either story is.