Distance between points along polyline ArcGIS for Desktop?

I'm using ArcGIS 10.3 (Desktop), and I'm trying to measure the distance between points following a path determined by a set of polylines. Hand measurement is not a viable option-- there are ~7500 points, and my polyline set is the global coastline. The ultimate goal is to get the spacing of these points along the polylines, so that I can analyze this with respect to a few other variables.

I can read python-- with a bit of effort-- but not write or edit it, so code-based solutions will not be particularly useful to me.

I have tried using the linear referencing tools-- I snapped the points to the lines, then transformed the polylines to routes and used 'locate features'-- but because my points are not (entirely) ordered along the path, using the difference between linear reference locations of subsequent points yields some fairly unlikely answers. In addition, spot checks of (hand-measured, using the built-in path length measuring tool) point-to-point along-polyline distance yield different answers than given by the corresponding linear references for the point pair. (Some of the answers were close enough that they could very well be measurement error on my part, but others were nearly 20%, despite my having zoomed in and snapped the measurement cursor to the polyline, proceeding vertex-to-vertex.)

Sorting first by linear distance seems to fix the first problem, but in light of the second one, I remain skeptical of this process.

I then tried simply splitting the polylines into segments ('split line at point') using the (snapped) points as the dividers, with the eventual goal of finding the segment length with the calculate geometry feature. This produced gibberish. I'm not sure what criteria ArcGIS used to split the lines, but it wasn't the point set I provided. Segments in the resulting shapefile ran right through tens of points, only to terminate at a place with no point. (Also, there were segments utterly empty of points...you get the gist.) I reiterate that the pointset was specifically snapped to the polylines, so this problem shouldn't be caused by excessive distance between the point and the polyline.

I'm at a bit of a loss, here: could someone point out where I might have gone astray and/or suggest a solution? Or, failing that, a new method that I could try? It seems like a fairly basic (not necessarily simple, but fundamental) analysis to do with geographic data, and I don't understand why it has presented so many difficulties.