What do you think about the discrepancy between BezierCurve and its DiscretizeGraphics version (this is where the question came from)?

Consider

pt = {{93.2759`, 277.0452`}, {90.6249`, 273.3252`}, {79.7499`, 
    255.70020000000002`}, {76.9999`, 250.70020000000002`}, {74.2499`, 
    245.70020000000002`}, {70.2499`, 237.70020000000002`}, {69.9999`, 
    235.32520000000002`}};

g = Graphics[{BezierCurve[pt]}];
Show[DiscretizeGraphics[g], g]

The discretized version doesn't quite match up. There is a small but consistent difference between the curves. Also, the bottom endpoint doesn't match. Bug?

Based on your example: This is a bug, no doubt. DiscretizeGraphics makes the exact same mistake I made. It simply uses BezierFunction with all the points form the BezierCurve, ending up with a single high degree function instead of a combination of several degree-3 ones.

pt = {{0, 0}, {0, 1}, {1, 1}, {1, 0}, {2, 0}};
g = Graphics[{BezierCurve[pt]}];
Show[g, ParametricPlot[BezierFunction[pt][x], {x, 0, 1}, 
  PlotStyle -> Orange], DiscretizeGraphics[g]]

Or do it yourself, the construction is quite easy:

Clear[Smoothed, CreateGraphics]
Smoothed[pts_List, \[Alpha]_?
   NumericQ] := {1 - \[Alpha], \[Alpha]}.# & /@ Partition[pts, 2, 1]
CreateGraphics[pts_List, \[Alpha]_] := 
 Module[{sublines, smooth, \[Theta], gr},
  sublines = NestList[Smoothed[#, \[Alpha]] &, pts, Length[pts] - 2];
  smooth = Nest[Smoothed[#, \[Theta]] &, pts, Length[pts] - 1];
  sublines = 
   Riffle[Line /@ sublines, Hue /@ Range[0, 1, 1/(Length[pts] - 1)]];
  gr = ParametricPlot[smooth, {\[Theta], 0, 1}];
  Show[{Graphics[{sublines, PointSize[Large], 
      Point[smooth /. \[Theta] -> \[Alpha]]}], gr}, 
   PlotRange -> {{-2, 3}, {-1, 2}}]
  ]

Manipulate[
 CreateGraphics[
  pt, \[Alpha]], {{pt, {{-1, 0}, {-0.8, 1.6}, {0.8, 1.8}, {1.8, 
     1.2}, {2, 0}, {1.4, -0.8}}}, Locator, 
  LocatorAutoCreate -> {3, 8}}, {{\[Alpha], 0.3}, 0, 1}]

giving:

Doesn't this do exactly the same thing as BezierFunction? I mean, it is still necessary to stitch together multiple degree-3 parts with this method too.

I think Bezier Curve by default creates a curve by combining several curves with SplineDegree 3. So, you can get your two plots to match by setting the SplineDegree of the BezierCurve to the number of control points.

But that would be getting your BezierCurve to match your BezierFunction, and you were asking for the other way. In that case, you'd have to replicate BezierCurve's default behavior. You'd need to create sever BezierFunctions by taking your control points 3 at a time. If you wanted to have one single function with parameter running from 0 to 1, you'd have to scale each individual BezierFunction to the appropriate part of the parameter's domain and compose them all together.