That's a pretty simplistic explanation. If you look at the details, the magnetosphere is protecting the atmosphere. Take each kind of radiation in turn.

X-Rays: The magnetosphere has no effect

UV: The magnetosphere has no effect

Cosmic rays: While both the Earth's magnetic field and the sun's magnetic deflect some cosmic rays, the Earth's atmosphere is mostly opaque to cosmic rays.

Solar wind/coronal mass ejections: The Earth's magnetic field does block these, but if it wasn't present, the atmosphere would block them (but of course be slowly stripped away in the process)

The fact that the magnetosphere does deflect some particles but isn't what makes life on Earth possible (other than protecting the atmosphere) is made most clear by the ISS being well within the magnetosphere but still receiving hundreds of times the amount on Earth.