It is worth mentioning that the Tutte embedding can produce good results for many of these graphs, e.g.

Graph[
 GraphData[{"SierpinskiCarpet", 3}, "EdgeList"],
 GraphLayout -> {"TutteEmbedding"},
 VertexSize -> 1/2
 ]

But it only works for 3-connected graphs.

And the implementation in Mathematica does not always produce ideal results:

Graph[
 GraphData[{"Apollonian", 5}, "EdgeList"],
 GraphLayout -> "TutteEmbedding"
 ]

Graph[
 GraphData[{"Fullerene", {60, 1}}, "EdgeList"],
 GraphLayout -> "TutteEmbedding"
 ]