Rights of way tended to be recognised and preserved by the original Act of Parliament that authorised the construction of the railway. In this case the relevant statute is the Inverness and Perth Junction Railway Act 1861.
A fairly detailed map would have been lodged with the statute that would have identified the relevant rights of way. Neither the statute nor the map is available online so far as I can tell. However, they should be available either at the Scottish National Archives or at the Archives at Kew (or at Network Rail in York I think).
Assuming that the crossing that is the subject of the original post was preserved as a public right of way (rather than a private one for the benefit of, say, the estate), then it can only be closed (whether permanently or temporarily) following a specified procedure. This would require prior notice and consultation. Given the apparent local reaction, this does not seem to have been done.
I have found reference to a Thieves’ Road, which is clearly regarded as a right of way, that is stated to run from Dalwhinnie to Corrour and beyond to Fort William.
https://www.railscot.co.uk/locations/D/Dalwhinnie/But the map found here indicates that this Road may have run from where the railway arch is:
http://www.heritagepaths.co.uk/mapsearch.php?path=226#zoom=13&lat=56.9129&lon=-4.2774Previous comment that a path does not have to be recorded as a Core Path for there to be a right of way is correct, including if there is an alleged gap in the Core Path at the relevant authority boundary. NB the Ordnance Survey map indicates that the boundary is to the west of the line.
The Cairngorm NPA Core Paths Plan (
https://cairngorms.co.uk/wp-content/uploads/2015/06/CorePathsPlanFinal150727.pdf) shows a Core Path as passing under the arch to the south of the station but not one over the crossing. This is may have been because Network Rail objected to the designation over its operational property.
But the starting point must be 1861 Act of Parliament. If that shows a public right of way at this crossing, Network Rail needs to be asked how it has been closed. If Scotways and/or the Ramblers are involved, they will probably know all of this.
Assuming a public right of way was preserved at the crossing back in the 19th century and still exists (and has not been temporarily closed), then crossing at this point in itself should not be an offence.