The Lost Mine of Phandelvar leans too heavily into the fight-'em-up genre of D&D campaign. The dungeons lack the design to properly facilitate a "dungeoncrawl," and the linear main plot and collection quest side plots are not peak role-play scenario.
The Lost Mine of Phandelvar is set up like a video game, and I don't think that's an ideal approach for a tabletop RPG. "Find specific NPC to give you a quest," and go from there into some monster-bloated dungeons, ultimately following the nominal low-level money rewards as motivation.
However, The Lost Mine of Phandelvar does provide a lot of good resources to expand upon. After an introduction goblin hideout venture and a free-town-from-bandit-mob-rule objective when the party first arrives in Phandalin, the world opens up. Phandalin hosts 9 name-specific NPCs, most of whom have both side quests to offer and some backstory that connects to the rest of the Sword Coast environment. For example, Halia Thornton at the Miner's Exchange offers a reward for the death and personal affects of Redbrand bandit leader Glasstaff, and she also is a member of the Zhentarim faction, to which she will extend an invite to the player characters on a job well done. The Zhentarim faction, a.k.a., "The Black Network," has a presence in Waterdeep, and its members have roles in the Waterdeep: Dragon Heist official campaign module. The train of thought is, The Lost Mine of Phandelvar is a robust addition to the Sword Coast setting.
Wizard of the Coast has put quite a bit of effort into placing a number of official campaign modules in settings up and down The Sword Coast. These include, but are not limited to: Storm Lord's Wrath, Waterdeep: Dragon Heist, Sleeping Dragon's Wake, Dragon of Icespire Peak, Divine Contention, Baldur's Gate: Descent into Avernus.
The Lost Mine of Phandelvar is compatible with the world and plots of all of these campaigns. But it takes some minor tweaks to do it effectively. I recently watched a Matt Colville video in which he describes his approach to a sandbox-style D&D campaign. The premise he gives is that he will set up the players in a neutral territory and introduce a bunch of story hooks, each appealing to a specific player's character and their motivations, and that the party is immediately put in a position of having to prioritize and strategize their queue of heroic exploits. Each of these hooks corresponds to a different low-level campaign module, the settings for which Colville places all accessibly within one general region.
I think that The Lost Mine of Phandelvar, in tandem with the other Sword Coast-located campaigns, would work very well to this effect. I would place a starting party in Phandalin, and there would be a number of directions to go:
Investigate the Redbrand Bandit problem and learn about The Black Spider and his bandits, Gundren Rockseeker, and Wave Echo Cave, in the vein of the canon The Lost Mine of Phandelvar.
Meet and become acquainted with Halia Thornton, probably link her to a separate throwaway side quest, after which she invites the party to join the Zhentarim. The party members go and man a hideout outside Waterdeep, encounter and kill a team of PC-proxy characters, and then conveniently pick up in the Waterdeep: Dragon Heist campaign at that point in Chapter 1 of the module.
Merchants and prospectors arrive in town, acting suspiciously, and through some investigation involving Elmar Barthen and Halia Thornton, the party learns about cultist spies and an impending attack on Phandalin (a la Storm Lord's Wrath). After surviving the attack, there merchants hire the party to escort them back to Leilon, from where a number of Storm Lord's Wrath and Sleeping Dragon's Wake quests are available.
Within the campaign module itself I have a handful other adjustments I'd make were I to run this campaign again. In addition to the above comments with regards to the hooks, world integration, and, ultimately, motivations, I would make some changes to the encounters. My earlier comment about the 'video game'-y-ness of The Lost Mine of Phandelvar applies in part to the encounters. Each dungeon/hideout/ruins/cavern map features a dozen or so rooms, each with a paltry set of goblins, undead, bugbears, indigenous monsters, etc. This makes no sense with regards to encounter design and made very slow pacing when the players got zealous for completionism.
Consolidate the encounters. Feature maybe half the number of encounters by combining groups of baddies. A lot of the maps include narrow hallways and cramped rooms. And while this is a useful mechanism to prevent lower level characters and newer players from getting overwhelmed, it also makes combat very one-dimensional, literally. My proposal for a simple change: initiate battles when the party enters any wider space, most of which connect to multiple rooms or hallways. Enemies file in from multiple directions, sometimes in waves, sometimes all at once, and the party has a bit more room to maneuver, implement tactics, and to experience more challenging, faster-paced combat. The party also won't get stuck exploring a dungeon for 3 sessions because of the time it takes to initiate and complete innumerable small bouts.
A number of these critiques and suggestions are based on my personal style and preferences as a DM. I struggle with running linear and combat-heavy scenarios and prefer a more open-world and more targeted, planned encounters. I want to make Phandalin into a living, changing environment, and I think that's very possible with these minor adjustments and with recourse to the expanded catalogue of official D&D modules (as well as, I'm sure, countless options over on DMsGuild/DriveThruRPG and via other such resources).