I'm inclined to think that the supplied-by-bearing-dealer grease is more probably one complying with MIL-PRF-81322 general-purpose aircraft grease than it is an automotive wheel-bearing-and-chassisj "red" grease. The first MIL-PRF-81322 grease that comes to mind is Mobilgrease 28, and it is indeed dyed a deep red.
Exxon/Mobil's product data sheet says this about industrial application of Mobilgrease 28: "Mobilgrease 28 is also recommended for industrial lubrication, including sealed or repackable ball and roller bearings wherever extreme temperature conditions, high speeds, or water washing resistance are factors. Typical industrial applications include conveyor bearings, small alternator bearings operating at temperatures near 177ºC (350ºF), high-speed miniature ball bearings, and bearing applications where oscillatory motion, and vibration create problems."
That said, the observed heating is much more likely to be caused by over-filling the bearings with grease or excessive pre-loading than by the differences between Mobil One and Mobilgrease 28.