The primary use case is to get results similar to warmer photosensitive papers used for B&W prints in the pre-inkjet past. By using a paper with such a color cast, monochromatic inks (various shades of gray and black) can be used rather than depending on color inks to give the image a warmer tone. This allows results to be much more consistent over time. (That is, I can print a copy of an image today that will likely be more consistent with an image I may have printed two or three years ago on a different printer.)

Additionally, wouldn't profiling bring the Warm tones back to neutral, with the printer trying to make up for the color cast?

Not really, because the "pure whites" in the image are areas that get no ink and rely on the color of the paper for how they appear. CMYK ink can only darken paper that is lighter than the inks, it can never brighten the paper. So while you might use a warmer paper and correct it to a neutral gray by adding some cyan ink, it would be a shade of gray, not white.