I can't answer the question about "place authorities" ignoring the leading part, but I think we should step back and consider the human reading the information first. It might help to note that I'm looking at this from the other side of the Atlantic, so I may not be well up on the assumed structures of US place names. Which is probably an important point if we want global clarity.
"Beacon Mountain, New York, United States" conveys meaning to me. It's a big hill!
"Fishkill, Dutchess County, New York, United States" also conveys meaning - but a different one. If you told me that a plane crashed in Fishkill, I'd imagine it hitting the houses of the town or coming very close. Looking at Google Maps, the crash site appears to be potentially 3 or 4 miles from the houses of Fishkill. This option appears to me to be misleading, therefore, since I read Fishkill as the settlement of that name, not as an administrative area of the same name.
"Beacon Mountain, New York, United States" or "Beacon Mountain, Dutchess County, New York, United States" (does that work?) seem to convey most information to an outsider.
I instinctively dislike using the Mountain as an address - addresses, to me, refer to buildings or streets, etc. Besides which, it doesn't resolve the issue of what goes into the place.
Re the question of "place authorities" containing only administrative places and not geographic features. (I'm putting it in quotes because I think we're taking about "place authority software" not the authorities themselves.) Frankly, I think the compilers of those databases had better sort their act out and include geographic features and settlements that may physically exist but not be administrative places. I suspect there has been no real thought on this - it just happens to be that lists of administrative places are easy to come by.
To insist on the use of administrative places creates huge anomalies where a settlement, perhaps of some size, known to all and sundry by a specific name, is not an administrative place. And as @TomH points out, many of us will have births and deaths at sea in our trees. Exactly what is the administrative place for those?
So - to return to the question. I would serve the human first by referring to "Beacon Mountain, New York, United States" or "Beacon Mountain, Dutchess County, New York, United States". (Well, actually, I'd probably write "New York State, USA" but I'm never sure what to do with NYC and the state of the same name.)