In the foreword to The Book of Lost Tales: Part 1, Christopher Tolkien briefly discusses this decision, saying that he believed his father to have ultimately abandoned the idea of Eriol/Ælfwine as a framing device:
[I]t is certainly debatable whether it was wise to publish in 1977 a version of the primary 'legendarium' standing on its own and claiming, as it were, to be self-explanatory. The published work has no 'framework', no suggestion of what it is and how (within the imagined world) it came to be. This I now think to have been an error.
The letter of 1963 quoted above shows my father pondering the mode in which the legends of the Elder Days might be presented. The original mode, that of The Book of Lost Tales, in which a man, Eriol, comes after a great voyage over the ocean to the island where the Elves dwell and learns their history from their own lips had (by degrees) fallen away. When my father died in 1973 'The Silmarillion' was in a characteristic state of disarray: the earlier parts much revised or largely rewritten, the concluding parts still as he had left them some twenty years before; but in the latest writing there is no trace or suggestion of any 'device' or 'framework' in which it was to be set. I think that in the end he concluded that nothing would serve, and no more would be said beyond an explanation of how (within the imagined world) it came to be recorded.
History of Middle-earth I The Book of Lost Tales Part 1 Foreword
We may argue over whether or not Christopher Tolkien was correct to believe this, but we can see a clear decline in the direct involvement of Eriol/Ælfwine over the development of the story; by the post-Lord of the Rings Silmarillion, he's been reduced from being an active participant in the story (as he was, as Eriol, in The Book of Lost Tales) to being a background character, with marginal notes and preambles only.
Christopher Tolkien notes Ælfwine's diminishing role later in Morgoth's Ring, following the long debate over the fate of Míriel:
In these writings is seen my father's preoccupation in the years following the publication of The Lord of the Rings with the philosophical aspects of the mythology and its systemisation [sic]. Of the deliberations of the Gods the sages of the Eldar preserved a record among the books of their law. How far away from these grave Doctors seems the 'horned moon' that rode over Ælfwine's ship off the costs of the Lonely Isla (11.321), as 'the long night of Faerie held on'!
Ælfwine is still present as communicator and commentator; but there have been great changes in Elfinesse.
History of Middle-earth X Morgoth's Ring 3: "The Later Quenta Silmarillion" Part 2 The Second Phase IV "Later Versions of the Story of Finwë and Míriel"
I've heard other fans speculate that Ælfwine was omitted to reduce the complexity of the work and, while I can't find any direct evidence to support that, it seems a plausible theory. By this point it's clear that Ælfwine is only tangentially connected to the narrative; in earlier versions of the story he served as a way for the reader to enter Tolkien's world (a role played by the hobbits in Lord of the Rings), but he no longer plays that role in later writings. Christopher Tolkien may have felt that disrupting the narrative with the occasional "quote Ælfwine" would have been needlessly disruptive to the reader, and I for one would have a hard time arguing with that.