Right, and there are Mongolian-like descriptions of Easterlings and Middle Eastern/African-like descriptions for the people of Southron and Harad. These are direct references to what these people looked like. So it would be sensible to visually adapt them to look as they are described.
How they dress, what they eat, what weapons they use and everything else about their culture that hasn't been explicitly described can be then creatively adapted to fill in the blanks. But representing them fully as a 'racially diverse' people wouldn't be filling in blanks, it'd be contrary to the descriptions we have that imply a certain look to them.
For example, having a racially diverse depiction of Southrons is perfectly fine for any visual adaptation, but may not be considered faithful or sensible to the source material.