That’s why they each carried radiation dosimeters, to record their exposure. While it is true shielding completely would be impossible or impractical, that doesn’t mean it will kill them. According to a transcript I heard, the medical people informed one of the flights they had received (at that point) the equivalent maximum daily radiation exposure of someone working in the nuclear industry. The Van Allen belt is not a place you want to hang around in, but it won’t kill you. In fact near as I can tell all of the flown Astronauts are pretty hardy stock and usually lived to a ripe old age. Jim Irwin was the first to pass, and from cancer iirc. Pete Conrad was killed in a motorcycle accident. But for the most part very healthy and sharp. Buzz Aldrin for example is still extant, and working on his 3rd wife.
If you do get around to watching and listening to the press conference and they start talking about the on board computer on the lunar module on the moon (I believe Aldrin does that) this is the link you need:
https://www.aulis.com/pascal.htm
It proves he was lying.