Can garbage be sent into the (near-)center of the world?
No. In fact, I don't know why David Hammen didn't say that instead of commenting.
The deepest hole that mankind ever dug was barely a scratch on the Earth's surface when you look at its size. They didn't even reach the mantle. The hole is only 12km deep, took about 20 years to create, and is still only 1/3 of the way to the mantle! Also, the hole is nothing through which you could dump garbage. It's only NINE inches in diameter.
Have you wondered what it means to dump trash into the Earth? Trash takes up space. There are no known voids in the deep Earth. It is very dense. To make room for trash, first you would have to dig the IMMENSELY expensive hole. Doing this also wastes energy. Then, you would need to invent technology to dig out a massive space, working from inside a nine-inch hole that goes over 7 miles into the Earth's surface. How expensive would that be? Is it even possible to carve out a space in the Earth under that much pressure without it collapsing? Given that scientists have only managed to keep open a nine-inch hole, you can bet we currently don't have available technology to build any kind of void for trash, let alone a void big enough to hold enough trash to make more room on the surface.