How to draw the outline of an icon made of several polygons?

With Mathematica 12 a possible approach is OuterPolygon:

pol = Polygon[{{{0, -1}, {1, 1}, {-1, 1}}, {{-1, -1}, {1, -1}, {1, 
     0}, {-1, 0}}}]
Graphics[{FaceForm[], EdgeForm[{Thick, Blue}], pol}]

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Graphics[{FaceForm[], EdgeForm[{Thick, Blue}], OuterPolygon[pol]}]

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You can use BoundaryDiscretizeGraphics:

bdg = BoundaryDiscretizeGraphics @ speakerIcon

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Graphics[{EdgeForm[{Black, Thin}], FaceForm[], MeshPrimitives[bdg, 2]}, ImageSize -> 30]

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Alternatively, you can use BoundaryDiscretizeRegion + RegionUnion:

bdr = BoundaryDiscretizeRegion[
  RegionUnion[Triangle[{{0, -1}, {1, 1}, {-1, 1}}], Rectangle[{-1, -1}, {1, 0}]]]

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