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I am trying to identify the closest driving/walking distance to major roads. I have settlements (points), the road from OSM (polylines), and major roads (polylines).

I have tried Network Analyst but it only works for few settlements because most of the settlement points are not located right on the OSM roads. I have tried Integration tool (Data Management) to put them on the roads but it messes up the settlements data.

What would be the easiest way to find the shortest driving/walking distance from settlement points to closest point of major roads' polylines?

I am using ArcGIS Desktop 10.3.

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You should increase the "search tolerance" in network analysis settings (screenshot). This value defines the maximum "snapping distance", in which points are snapped to your road network.

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