It's unlikely QGIS (or any software) has implemented the exact formulas provided in the antecedent thread, but one could sample a polyline at regular intervals, compute the distances between the sample points and the "node," and average them: these are common operations available in many GISes. (This is a Riemann Sum, or rectangular, approximation to the integral. Improved approximations, such as with the Trapezoidal Rule or Simpson's Rule, can be implemented similarly. Correct application of Simpson's Rule would require estimating the average distances separately over each segment of the polyline and then forming the segment-length-weighted average of those results.)
This screenshot (using ArcView 3, an old simple GIS :-) illustrates the procedure. A 10 km feature in a road layer (barely visible in black) has been sampled at 100 m intervals (shown in darker cyan) beginning at a random location. Another point layer containing a single point (shown as a cross) has been spatially joined to the sample points. One result of the join is to compute a [distance] value to each of the sample points, shown in the table at left. A statistics dialog displays the mean distance (of 3503 meters), shown highlighted. A circle of that radius (red) was manually added based on that information.