[This is Part 1 of the Answer]
This is a very complex question that depends on understanding Classical allusions and Christian Biblical allusions. I think we've found the answer with a little (or a lot...) "thinking out loud."
If Milton is to be trusted, then "Lycidas" certainly is about a larger issue as well as being a memorial to his shipwrecked and drowned friend. In his introducing headnote, Milton says the monody "bewails a learned Firend" and additionally foretells of the ruin of England's corrupted Clergy, who were at the height of their power: "And by occasion fortels the ruine of our corrupted Clergy then in their height." Note that the headnote didn't appear in the original but was added afterward during Milton's significant revisions to the text after 1638.
The task now is to identify the part or parts that relate, undoubtedly in a metaphorical sense, to the "corrupted Clergy," and any other greater themes Milton might be addressing , though one must not overlook the fact that the poem is first a memorial to Edward King ( -1637) and only secon, by virtue of Milton's agile mind, a vehicle for the secondary discussion of the Clergy.
"Lycidas" is described by Milton as a monody, which is a poem for a single voice, or speaker. This troubles critics because two individuals are quoted within the text and a third invoked at the end, indicating to some that it is actually polyphonic, having more than one voice. It appears most logical that Milton meant "monody" in the sense of a story teller who might quote dialogue of several or even many characters, yet still be only one voice, one narrator, telling a story, which is quite a familiar narratorial device in novels that developed later. Another explanation for the monody label Milton assigns is that the story told by the first speaker is actually a recitation of the raconteur who Milton calls "the uncouth Swain" who appears in the last stanza, a technique used from time to time in later literature, in which a narrator revealed at the end has been quoting another narrator telling the tale.
The first stanza of Milton's monody, "Lycidas," sees the speaker calling upon the classic Greek gods as he "disturbs their season due" because he must "sing" [used in the sense that poets are classically called singers and were most often, if not always, accompanied by the lyre] the early passing of Lycidas. In the second stanza, he hopes that someone will do the same for him at the time of his "destin'd Urn."
In Stanza 3, the speaker describes how Lycidas and he had awaken and worked fields together. This is an important stanza to the question at hand as it is here that Milton introduces a Shepherd conceit in a Biblical allusion to Jesus as the great Shepherd whose believers and followers are his "flock": "Batt'ning our flocks...." Stanza 4 elaborates on the conceit and deftly weaves together a lament for Lycidas with a Biblical allusion to a lament for the slain Jesus, who is symbolized in the Bible as the Vine and the Rose, as well as the Shepherd. The last line laments that Lycidas' "song" of Line 36 has ceased.
In Stanza 5 the speaker wonders where all the powers were that might have saved Lycidas. It may be argued that another allusion to Jesus in his designation as "the Lord" appears in Lines 58-60 in the form of a comparison between Orpheus and Jesus. Orpheus is reckoned the son of Apollo, the Greek Sun god, and is called the greatest musician and poet, or singer, and a Biblical metaphor equates "the Lord" with "song": "The Lord is my strength and my song..." (Exodus 15:2).
The next two stanzas, nine and ten, return to lamenting Lycidas' death, bestowing a wealth of flowers on him and, in a final return to the Shepherd allusion, assuring that though Lycidas has "sunk low," his is "mounted high" and "is not dead" because "Through the dear might of him that walk'd the waves" (173), he will be risen, as his "oozy Lock's he laves," to go to the "blest Kingdoms meek of joy and love." The words "him that walk'd the waves" is another allusion to Jesus who is reported to have walked on the waters of the Sea of Galilee. This stanza completes the salvation and redemption theme developed earlier with a final allusion to Jesus in his prophetic role as "the Sovereign Lord" who "will wipe away the tears from all faces" (Isaiah 25:8).
Milton ends with "Thus sang the uncouth Swain...." There are several opinions on the identity of this individual, but it is logically probable that it is Milton's voice as the primary narrator who has been reciting a story originally told by another unknown or unnamed narrator, in essence creating what in literary novels will be a framing device.
In conclusion, "Lycidas" is about a greater idea than a lament for Edward King. In fact, it is about two greater ideas. The first idea is that "Lycidas" is a conceit for Christian redemption and salvation. The second is that the digression is a tongue-lashing for the Clergy whom Milton (in the voice used in the final Stanza) styles as false Shepherds who will be smitten with the Lord's double-edged sword.
[Information for this three-part Answer was taken from Lycidas, Milton Reading Room, Dartmouth College.]
[This is Part 2 of the Answer]
...In the above allusion, it may be argued that the conceit continues through the designation of Orpheus as the "inchanting son / Who Universal nature did lament..." as it is said that the universe lamented Jesus' death.
The sixth stanza covers three concepts beginning with the lament that it is thankless to be a lowly shepherd, accompanied by the suggestion that it might be better to frolick like others do with the Nymphs. The Stanza continues by saying the the pleasures or rewards ("Guerdon") thus expected are cut short by the "Fury" who's "shears" cut the silken thread of life: "slits the thin spun life." Finally, Phoebus, a by-name for Apollo who is earlier equated with Jesus as the Lord, encourages the speaker by saying that mortal "Fame" and "Guerdon" are not the goal but that the goal is the judgment of Jove who "pronounces lastly on each deed."
Stanza 7 adds more lament through Classical allusion to Lycidas demise. Stanza 8 introduces the discussion promised in the headnote sentence; "And by...Height." Stanza 8 is styled as a digression (digressio) from the logical order of the story and lament. "The Pilot of the Galilean lake / Two massy Keyes..." is a reference to Saint Peter who is reckoned in Christian tradition as the Keeper of the Keys to Heaven. He is also acknowledged to be the founder of the Christian Church. Therefore, he metaphorically represents the church at large and/or the Clergy of the Church at large. Along with introducing this metaphor, Milton returns in this Stanza to the Shepherd allusion with "fold" in Line 115.
In this digression, which is built on a metaphor and the continuing Shepherd conceit, Milton describes the Clergy as not caring about anything (116) other than scrambling for a good place at the spring feast for the shearers (117), even to the extent that they "shove away the worthy bidden guest," which also implies that the Clergy may not have been invited to the feast (118).
Milton accuses these individuals, the Clergy of the headnote, of being "blind mouthes!", and nothing more, who don't know how to shepherd sheep, even to the point of not knowing how to hold the shepherds most valuable tool for saving stray sheep, the "Sheep-hook." He further accuses them of knowing nothing of the "Herdsmans art," which is another allusion to Jesus.
Milton derides these no-nothing imitation shepherds saying it doesn't concern them that they know nothing as they have what they need: "What recks it them? What need they? They are sped;..." Milton contrasts this with the sheep who are hungry and in pain and neglected and at risk of attack by the "Woolf." Lines 130 and 131 assert that at the door stands someone holding a double-edged ("two handed") sword who will "smite once" and be so effective in smiting the offending Clergy that only one "smite" will be needed. This is a Christian allusion to the mighty word of God as a double-edged sword: "For the word of God is living and active. Sharper than any double-edged sword." (Hebrews 4:12).