How do I find a point at a given distance along a route? epoly.js is giving me extremely inaccurate and unusable results

I’m trying to find the point along a Google Maps API Directions route given a distance from the starting point. I have my code working and it gives me very accurate results most of the time. However when I make very long directions requests (for example, 1,000+ kilometers), the results are less accurate, and the longer the directions route the more inaccurate the results are. Once I reach approximately 3,000 kilometers the results are off by about 4,000 meters, which is entirely unacceptable for my application.

The function I’m using to compute the point is from epoly.js, and the code is as follows:

google.maps.Polyline.prototype.GetPointAtDistance = function(metres) {
    if (metres == 0) return this.getPath().getAt(0);
    if (metres < 0) return null;
    if (this.getPath().getLength() < 2) return null;
    var dist=0;
    var olddist=0;
    for (var i=1; (i < this.getPath().getLength() && dist < metres); i++) {
        olddist = dist;
        dist += this.getPath().getAt(i).distanceFrom(this.getPath().getAt(i-1));
    }
    if (dist < metres) {
        return null;
    }
    var p1= this.getPath().getAt(i-2);
    var p2= this.getPath().getAt(i-1);
    var m = (metres-olddist)/(dist-olddist);
    return new google.maps.LatLng( p1.lat() + (p2.lat()-p1.lat())*m, p1.lng() + (p2.lng()-p1.lng())*m);
}

What is causing the result to be so inaccurate over large distances and what can I do to make it more accurate? (I need accuracy down to approximately 1-3 meters) Does the Google Maps API have any way to do this or is epoly.js my best bet? If so, then what can I change about the above code to make it give more accurate results?

I’ve been searching for an answer to this for a while, but everything I can find either recommends epoly.js or shows code snippets that perform exactly the same computations as epoly.js. It seems as though Google doesn’t have any built-in way to do this, however I’ve seen something similar done in some applications like https://routeview.org, where you can clearly see the orange man tracing along the route perfectly even when navigating thousands of kilometers at a time, so I have to believe a higher level of accuracy is possible.

Here are two screenshots illustrating a short distance request and a long distance request. Note that it’s very accurate over short distances but becomes wildly inaccurate over longer distances. Also note that the marker in the second image is being viewed from far away. It may look close to the path, but it’s actually about 5,000 meters away on the other side of a large hill. (The marker in the first image is viewed from very close up, and even when viewed so closely it doesn’t deviate from the path any noticeable amount)

This image is for a 20km route:

This image is for a 3326km route: