Why does exhaust flow inwards through a radial turbocharger?
This has to fight against centrifugal force instead of leveraging it.

In fact, the opposite is true. The arrangement shown actually helps to extract energy, perhaps counterintuitively. A typical example of this arrangement is the Francis turbine.
A high flow speed plus high tangential velocity input, fed through to a low flow speed/low tangential velocity exit, will create the maximum energy differential for extraction. And that describes inwards flow.
There are a few ways of looking at this. One is to look at the exit velocity of the fluid. In a compressor, it's okay if the fluid has some leftover velocity as it will by converted to static pressure when slowed down by Bernoulli's law. For a turbine, the exit velocity should be as low as possible, as kinetic energy of the exhaust flow is 'wasted' energy. So it makes sense to place the exit at the inner radius of the turbine, where the radial velocity component is lowest.
Another way of looking at it is pressure. Centrifugal force creates a pressure gradient across the turbine in radial direction. As a 'packet' of air moves inwards, its pressure becomes lower. By conservation of energy, this must mean that it is expending work on its surroundings. The turbine extracts this work done from the packet of air moving inside.
Another intuitive explanation involves an ice skater retracting their arms in a pirouette (or a bored office worker on a swivelling chair). By conservation of angular momentum, their angular velocity increases. A packet of air would 'like' to increase its angular velocity, but this is kept fixed by the turbine rotating at a fixed velocity. The packet of air must thus be slowed down moving inwards, and as a result exerts a pressure on the turbine blade in front of it.
The choice between axial, centrifugal or mixed-flow is done based on the flow conditions, with high-flow low pressure differential favouring axial design and lower flow or higher pressure differential favouring the radial setup as shown. A more compact design means fewer turbine or compressor stages, which results in a higher pressure differential across the single stage. Thus, a radial setup is preferred.