Still works here with HDRI on a SKy object, image in the Diffuse channel and set to Multiply with the base Diffuse colour.
One thing of course is this may change the intensity of the HDRI, so you may need to adjust the Threshold of the bloom and glare (which only work off how bright any given point in the image is). Haven't tested but as well as simply "darkening" the HDRI, it could also be that you lose the high dynamic range in the HDRI by multiplying it with the base colour in the Diffuse. But as such, it doesn't just stop working, just your Threshold has changed :)