I think you are imagining the earth as one giant rigid inertial frame and that creates problems.
Let's look at the atmosphere, a giant doldrum over the pole to make it simple.
What keeps the air up there a certain height. Well there is a stronger pressure from the air below it than from the air above it.
Newtonian gravity would say the air stays at rest because the pressure gives a net force upwards that opposes the force of gravity on that parcel of air.
But GR instead says that there is a net force in the air so it accelerates upwards, which takes it to a region of spacetime where the metric looks the same and where there is exactly the same pressure imbalance.
This is a completely different story, but much more accurate. If there is mire pressure from below than from above and no other forces then you go up.
And that happens around the south pole and the north pole each parcel of air actually accelerates up because each parcel of air has more pressure from below than from above.
And that's why you don't fall to the center if the earth, the ground below you is in a compressed state and exerts pressure on your shoes which themselves are compressed and exert pressure on your feet and your keg is compressed and exerts a pressure in you midsection and so on. You are accelerating upwards in an inertial frame. Its just that the things around you are accelerating upwards to.
From your perspective the inertial frame is accelerating downwards. But the earth (the ground) is not moving inertially, the top layer of the earth is also feeling more pressure from below it than it feels from above and the rocks and such below you are also accelerating upwards.
So the problem is that you simply didn't let go of the idea of the earth as a rigid object that determines a frame. Each little infinitesimal patch of spacetime is an inertial frame and to you they look like they are accelerating downwards, unless you happen to be moving inertially. Which is uncommon.
Instead of infinitesimal inertial frames you can work with a finite coordinate neighborhood but then you need to use the curved geometry of GR.
So people are not being pulled to the center of the earth. The people and the parts of the earth are generally being accelerated (acceleration as always relative to an inertial frame) outwards because of things like pressure gradients. When they don't feel pressure gradients and such, they move inertially which might loom like acceleration to people that aren't moving inertially.
And if people standing on the earth look like they care accelerating in different directions on opposite sides of the earth it is just because each is experiencing pressure gradients pushing them away from the center because that's how the matter around them is arranged.