I'll give you one of the best "secrets" that I learned throughout the years after going through many different pains (leeching scripts, manual clicking, etc). It is an old-school trick... contact a real person there!
The best way do get the NED elevation dataset is to contact USGS's Eros group directly at bulkdatainfo_at_usgs.gov
You send them an external drive and after 4 to 8 weeks (usually much less than that) they will send you the entire dataset that you requested.
Then use GDAL to query your points in a way similar to this example. You may want to read the Affine Geotransform section of the GDAL Data Model