function to draw a 3-point arc

Simplification

fun[p_] := 
 Module[{cs = List @@ Circumsphere[p], c, r, u, v, w, reg, rm, 
   regc}, {c, r} = cs;
  {u, v, w} = Sort[Mod[ArcTan @@@ (# - c & /@ p), 2 Pi]];
  reg = {{u, w}, {v, 2 Pi + u}, {w, 2 Pi + v}};
  rm = Abs[Subtract @@@ reg];
  regc = Pick[reg, # == Min[rm] & /@ rm];
  ParametricPlot[c + r {Cos[t], Sin[t]}, {t, ##}, 
     Epilog -> {PointSize[0.03], Red, Point[p], Green, Point[c], Gray,
        Dashed, Circle[c, r]}, AspectRatio -> Automatic, 
     PlotRange -> Table[{-2, 2}, 2], Frame -> True, Axes -> False] & @@@
    regc]

Original Answer

Not efficient but for what it's worth (using Circumsphere):

func[p_] := 
 Module[{cs = List @@ Circumsphere[p], c, r, u, v, w, reg, rm, regc},
  {c, r} = cs;
  {u, v, w} = Sort[Mod[ArcTan @@@ (# - c & /@ p), 2 Pi]];
  reg = ParametricRegion[c + r {Cos[t], Sin[t]}, {{t, ##}}] & @@@ {{u,
       w}, {v, 2 Pi + u}, {w, 2 Pi + v}};
  rm = RegionMeasure /@ reg;
  regc = Pick[reg, # == Min[rm] & /@ rm];
  RegionPlot[regc, 
   Epilog -> {PointSize[0.03], Red, Point[p], Green, Point[c], Gray, 
     Dashed, Circle[c, r]}, AspectRatio -> Automatic, 
   PlotRange -> Table[{-2, 2}, 2]]]

Testing on 10 random triples:

pts = RandomReal[1, {10, 3, 2}];
anim=func/@pts;

Updated based on ubpdqn's brilliant use of Circumsphere. The function is now more than two orders of magnitude faster.

Ok, since it seems like there is no simple form solution based on additional readings of other posts, I am posting the general form of the function I suggested in the question:

arc3v2[pts_List] := 
   Module[{center, radius, theta},
     {center, radius} = List @@ Circumsphere[pts];
     theta = SortBy[Mod[ArcTan @@ (# - center), 2 Pi] & /@ pts, # &];
     {center, radius, {First@theta, Last@theta}}
    ]

Verifying that it works (the function in the original question didn't correctly calculate angles, as VectorAngle is not accounting for quadrants):

pts = {{1, 5}, {-1, .1}, {-.7, -0.1}};
arc = arc3v2[pts];
Show[
  Sequence @@ Table[Graphics[{Red, Disk[pts[[i]], 0.1]}], {i, 3}],
  Graphics[{Green, Disk[arc[[1]], 0.1]}], 
  Graphics[{LightGray, Circle[Sequence @@ Most@arc]}],
  Graphics @ Circle[Sequence @@ arc], 
 Axes -> True]

Comments are welcome!