DOES the engine also charge the traction battery?

Part of the power from the internal combustion engine (ICE) goes directly to the tires, and part of it is routed through the two motor generators (MG1 & MG2). Note that I said TWO MGs; the MFD display only shows one. The amount of power routed through the MGs in relation to the direct power is what determines the effective gear ratio, since the Prius lacks an ordinary transmission. The high voltage battery (HV) serves as both a temporary storage place for excess electricity and as a reservoir for added power from the MGs. When it needs to be charged, some of the energy from the ICE is used for charging. Even when it's not being charged you will see a small amount of current flowing back and forth between the MGs and the HV battery. The short answer to your question is that, yes, it's inefficient, but not as inefficient as an ordinary transmission. If you can figure out how to drive without speeding up, slowing down, or climbing hills, then you can do away with energy wasting transmissions.

Tom