I think you have two problems:
Non-symmetric control points
Initially you start with equal distances between p0 to p1 and p1 to p2. If the tolerance angle between the line segments is not met, you move p1 and p2 forward, but keep p0 where it was. This increases the distance between p0 to p1 while keeping the distance between p1 to p2 the same. When you create a curve using p1 as the control points, it can be heavily biased towards p2 depending on how many iterations have passed since the last curve. If you would move p2 twice the amount than p1, you would get even distances between the points.
Quadratic curves
As mentioned in other answers as well, quadratic curve is not very good for this case. Adjacent curves you create should share a control point and a tangent. When your input data is just points, Catmull-Rom Spline is a good choice for that purpose. It's a cubic Hermite curve, where the tangents for the control points are calculated from previous and next points.
The Path API in Android supports Bézier curves, which are a little different than Hermite curves regarding parameters. Fortunately Hermite curves can be converted to Bézier curves. Here is the first example code I found when Googling. This Stackoverflow answer also seems to give the formula.
You also mentioned the problem of sharp edges. With the input data you have, it's impossible to detect if there is an actual sharp corner or just a very steep curve. If this becomes a problem, you can make the iteration more adaptive by increasing / decreasing the step on-the-fly as needed.
Edit:
After further thinking quadratic curves could be used after all. Instead of drawing a quadratic curve from p0 to p2 using p1 as the control point, draw it from p0 to p1 using a new point p0_1 as the control points. See the picture below.
If p0_1 is in the intersection of the tangents in p0 and p1, the result should be smooth. Even better, since PathMeasure.getPosTan() returns also tangent as the third parameter, you can use actual accurate tangents instead of approximations from adjacent points. With this approach you need less changes to your existing solution.