“The earth’s magnetic field is very weak,”
Funny thing about Magnetic Fields - they decay at about 1/r^3 power.
That the earth has a field of about 0.4 G at 4000 miles, means it’s pretty strong at it’s source. CERN at a mile away is probably negligible compared to the earths at 4000 miles. CERN magnets are intended to deflect charged particle beams. They are strong if you are next to them, not so much if you take a dozen steps away.
On the surface of the earth the magnetic field is only about 0.6 to 0.65 G due to geographical variation. I will have to calculate what is the magnetic field strength at the core, I never thought of doing that calculation. Even small magnets can over come the earth’s field at the surface.