At current the Master Sword and Triforce are about the only things mandating a link between LttP and the OoT branches. Most story points these days may be seen as a continuation of the Vaati arc, particularly Ganon himself, seeings how OoT Ganon is likely dead in both forks and FSA provides the origin of trident-wielding Ganon.
Whilst it is true that the FSA provides many characteristic elements necessary to transition comfortably into LttP's story, it is the plot in the game's manual that absolute mandates some connection to Ocarina as well. After all, Ocarina's story was based on that very legend which introduces elements such as the Master Sword, the Triforce, the Sacred Realm and Dark World and especially the sages seal. All of these elements being present in some fashion in FSA, but removed entirely from focus as theirs is of little relevance to that game's plot.
Of course, LttP itself admits that these very legends it tells of are in no way fact, word of mouth changing facts and time obscuring the boundary between fact and fiction. Ocarina is but such a small aspect of the underlying truth. Too bad that underlying truth also has many contradictions to address, such as at the same time having to account for both the disappearance of the sea Zoras and the existence of the acient Hylian structures. Which strangely mandate branching out both ways from Ocarina.
the original LoZ has left us with one more Ganon death than we can account for, and no effort has been made to address that;
That depends, if Miyamoto´s mentioned order is correct; the Silver Arrows leave Ganon in quite a predicament. However, from simply LttP beyond, there are the concepts from Ancient Stone Tablets, which has no problem with making Ganon seemingly self reviving or otherwise some strange side effect of Link's wish on the Triforce. Though the "actual" events of AST hardly fit the continuity because of the Satellaview mascot, its story does give us some insight into the minds of the developers.
Overall, the original game's backstory has remained very self-contained, with no other title proving itself relevant to its setup.
Within the overall Hylian lore, the state of the Triforce is a fairly important matter that needs some setting up, which preciously little titles could account for. As for LoZ/AoL's effect on the other games; somehow or other the sleeping Zelda myth needs to find its way back into the series.