After much puzzlement, I noticed that my HeatExchanger:FluidtoFluid, set to control type CoolingSetpointOnOffWithComponentOverride is preventing the integrated chiller to turn on (at all) if you use a Minimum Temperature Difference to Activate Heat Exchanger that is greater than zero (and it won't accept less than zero...)

Also the HX doesn't seem to do much in this case either (but that could just be for a reason...), so I'm not that concerned with it.

Why doesn't the HX let the chiller run when it clearly cannot meet the load itself?

You can take the E+ FreeCoolingChiller.idf example file and change the Min temp of the HX to something like 1C or 2C, and see for yourself.