One can use Iterator in the "GeneralUtilities`" package to advance through the colors:
Needs@"GeneralUtilities`";
cyclicListIterator[input_List] :=
GeneralUtilities`NewIterator[cyclicListIterator,
{i = 0, n = Length[input], list = input}, list[[Mod[++i, n, 1]]]]
colors = "DefaultPlotStyle" /. (Method /.
Charting`ResolvePlotTheme[Automatic, ListPointPlot3D]) //
cyclicListIterator;
ListPointPlot3D[{#}, PlotStyle -> Read[colors]] & /@ list // Show

Note: The Iterator[] for the colors needs to be reset after each use. One could use With[{colors = ...}, ListPointPlot3D...].
The same thing but with old-style MapThread:
colors = "DefaultPlotStyle" /. (Method /.
Charting`ResolvePlotTheme[Automatic, ListPointPlot3D]);
MapThread[
ListPointPlot3D[{#}, PlotStyle -> #2] &,
{list, PadRight[colors, Length@list, colors]}
] // Show
(* same figure as immediately above *)