As much as any adventure requires the players to have a bond with the local community. Such a bond can help players get invested in the adventure, because it can provide a motivation to get involved. If your players are more the sort to engage in whatever plot falls in front of them because it’s there, then bonding with the community only matters inasmuch as it makes for a more engaging story.
The Shattered Obelisk portion of Phandelver and Below starts with the players getting back to Phandalin from Wave Echo Cave to find that while they were gone, some goblins attacked the town and caused some mischief, stealing some strange items, doing some vandalism, etc. and the locals want help repairing the damage and recovering what was stolen - which turns out to have been fragments of one of the mysterious obelisk, which had unknowingly been used as construction materials. Everything else in the module kind of follows from there, with the specific faction of goblins that stole these fragments being the initial antagonists, though they later turn out to be minions of the actual main antagonists. It’s not even the same faction the PCs deal with in Lost Mine.
Shattered Obelisk really has nothing more to do with Lost Mine than being set in the same location, and a vague expectation that the PCs know and are known by the Phandalin locals. The hardcover version kind of tries to retroactively inject some foreshadowing to Lost Mine by adding rumors of a few goblins with “strange” weapons being spotted among the cragmaws. It’s very clumsily tacked on, in my opinion.
