[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).