From my point of view this result was not in line with the explanation Safari assigns the display profile to the untagged file as in the case of the sRGB display this shouldnt result in a red colour cast if the profile of the sRGB device would have been assigned.
Some time ago there was talk of an intermittent OS X bug where the same profile would be used for both monitors (even if independently calibrated). Maybe that could explain this.
In all color management operations there is a source profile, and a destination profile. One is converted into the other. If the two are identical, color management is effectively turned off because no profile conversion occurs. Zero sum. So assigning the display profile to the file is another way of saying that this file is not color managed.
The only browser usable with a wide gamut monitor is Firefox. It can be set to assign sRGB to untagged material, so that the color management chain can operate.
Many websites strip the embedded profile.