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.

One of the flaws with using subduction zones as dumps for toxic chemicals so they can be broken down by heat and pressure at depth is the rate of sinking of the subducting plate.

Subduction is a very slow process, generally between 2 and 8 centimeters per year. Any toxic material is going to hang around the interface for a very long time, with the potential to contaminate the sea bed and farther beyond, through transportation via submarine currents, for a very long time.

Also, depending on the subduction zone used, the toxic material may resurface relatively quickly, particularly where island arcs are created. This would contaminate the islands made in such environments.

Your idea may seem weird, but it is what has been done for decades in the US and elsewhere.

You do not need to reach the centre of the earth. 3 to 4 km may suffice (Terms and Conditions apply ...)

Have a look for example at the Rocky Mountain Arsenal disposal experience in Colorado USA). In the 60s, the US military had to dispose tons of chemical and batteriological byproducts from their production. You may find details about the site here https://en.wikipedia.org/wiki/Rocky_Mountain_Arsenal but what makes it interesting regarding your question is that the deep disposal of fluids led to ... earthquakes. See relevant publication from USGS archives: https://earthquake.usgs.gov/static/lfs/research/induced/Healy-et-al-1968-Science-(New-York-NY).pdf

Denver area was (and still is) expected to be tectonically quiet, with no earthquakes, and there are convincing proof that the disposal of fluids (toxic, heavy metals, batteriological byproducts, who know what they did dispose) induced some strong earthquakes up there.