You raise many valid points. However, the Sierra Club did not originate the idea of public land being held in trust by the government, that was original to the foundation of the country. I had a Constitutional Law class many, many years ago where that specific topic was dealt with at length in discussion of the Tenth Amendment.
I cited a Wiki link because it was quick and easily accessible, kind of like you cited a BLM link (was that you who did that?).
You are correct, but the public trust doctrine has historically only been applied in the United States for seashores and land underneath navigable waters, i.e. the river beds. It has not be applied to unallocated lands, although that is now the claim regarding the FLPMA. That was certainly not the legislative intent of the FLPMA.
The things I posted were not BLM documents, but Congressional Research Service papers. I use Wiki as well, but I have been burned, so I exercise great care.