Evolution of the (phenotypic) facial features of the indigenous people of (west/central) Africa
Note that I had never heard of the expression "broad african type" and could not find much reference to this term when googling. But anyway, I'm happy to use it to refer to a particular set of typical facial features.


  What is the reason behind this seeming resemblance? 


Note that the Homo Heidelbergensis image is not a picture (obviously) and much of what is being depicted was open to artistic interpretation. It is possible that some of the feature you are thinking of are not real. Have a look at the post In reconstructions, how are various shapes of facial features determined from skull only?.

You list several phenotypic traits and it would be too broad of a question if you were expecting an explanation about the evolution of each of these traits. I will just pick one classic trait; the nose.

One of the role of the nose is too even up the temperature of the air that we're inhaling. As such, the shape of the nose is under differential selection pressures depending on the environment. So if the nose of Homo Heidelbergensis looked alike (assuming it really looked like on the facial reconstruction) today's "broad African type", it is probably because of shared environment. Note however that nose shape vary a lot among african lineages and also among non-african lineages. You might want to have a look at this popular article.

A big point of ressemblance is the skin color that you don't talk about. Have a look at the posts Why do we assume that the first humans were dark-skinned? and How did some humans evolve to be white? to learn more about skin color


  Is this appearance an illusion and possibly a construct of a prejudiced mind, or does this have a valid scientific answer?


First, see my above point about facial reconstruction.

H. Heidelbergensis lived in Africa and in this resepct shared an environment that is like the environment that modern african are encountering that what modern Han Chinese are encountering. In this respect, we could expect some ressemblance. 


  For example, are most Africans closer to our ancestors than the rest of humanity is?


No, they're not and they cannot be. The MRCA of all humans today is more recent than the MRCA between any human and H. Heidelbergensis. You should have a look at this intro to phylogeny to understand a little bit better how different lineages relate evolutionarily speaking.