Nested list to graph

edges = Cases[li,
   {node_String, subtrees__List} :> (
     node \[DirectedEdge] #[[1]] & /@ {subtrees}),
   {0, ∞}] // Flatten

Note the level specification within Cases.

Graph[edges,
 VertexLabels -> "Name",
 ImagePadding -> 30,
 GraphLayout -> {
   "LayeredEmbedding",
    "RootVertex" -> "fig"}]

With IGraph/M running on Mathematica 11.3,

IGExpressionTree[li /. List -> Construct]

enter image description here

li //. {{x_, rest__} :> x[rest], {x_} :> x} // 
 TreeForm[#, DirectedEdges -> True] &

enter image description here

A similar rule can be used to parse JSON data and display with TreeForm

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