Poetry and the War
It should be noted, first, that the war produced an absurd number of poets at every level -- not just in the trenches. Catherine Reilly's groundbreaking English Poetry of the First World War: A Bibliography (1978), for example, lists some 2,225 English poets publishing work about the war while it was going on, and soldier-poets were far from the majority -- 1,808 of this number were civilians. A quarter of the aggregate number were women, too, which is another fact often overlooked; it was only with a strong feminist turn in First World War studies in the 1970s that these many excellent poets were finally given their due. Prior to that, it was often very difficult to find their work at all unless they were power-house figures like Jessie Pope or Mary Borden, and even then it could be difficult; Brian Gardner's Up the Line to Death: The War Poets 1914-1918 (1964), for example, set a bold new direction for the canon of the war's poetry, but among its seventy-two included poets (many justly being rescued from obscurity) there isn't a single woman. Anyway, this is something of a sideline to our main subject -- sorry.
It was popularly felt from the very start, at least in Britain, that the war demanded an extensive poetic response. We'll get to why that was in a moment, but it would first be worth examining just what that response looked like. In August of 1914, for example, the Times of London received upwards of a hundred submitted poems a day from its readers, with around a thousand finally having been published by the time the year was out. Anthologies of war-related poetry began coming out at once, often for the support of various charitable endeavours (such as the relief of Belgian refugees), and patriotic poetry readings were a popular public entertainment for young and old again. Sales were excellent, too; Kipling's Barrack-Room Ballads (first published in 1892) sold 29,000 copies in 1915 alone, and by the war's end the complete poetical works of Rupert Brooke -- then the most celebrated martyr from among the soldier-poets -- had sold some 200,000 copies.
It may surprise you to hear it, but the consolidation of a coherent poetic response to the war at the highest levels was one of the first tasks of Britain's first-ever official War Propaganda Bureau, based at Wellington House in Buckingham Gate. The Bureau put out a call as soon as the war was declared soliciting the immediate involvement of the biggest names in English literature, and on September 2nd of 1914 those figures met in what must have been one of the largest such gatherings of establishment literary figures in history. Everyone was there, or at least sent a representative -- Rudyard Kipling, Thomas Hardy, Sir Owen Seaman, G.K. Chesterton, Sir Henry Newbolt, the poet laureate Robert Bridges, the far-future poet laureate John Masefield, and many others besides (to say nothing of major figures known now mainly for their prose, like H.G. Wells, J.M. Barrie or Arnold Bennett). I could go on about all of this forever, anyway; interested parties can read my recent piece on the subject at Oxford's WWI centenary project, Continuations and Beginnings.
Georgian Poetry and the Post-Victorian Cultural Milieu
Now, you've asked why there were so many soldier-poets, and suggested the role of education in the matter. This is a good start, and helps explain some of the enormous popularity of the poetic genre on a broader level. The passage of the Elementary Education Act of 1870 had seen the establishment of rigorous new standards for education throughout the realm, and one of the consequences of this was a significant boom in mass literacy. By the time the war began the first children to be subjected to the standards of the act were well into their adulthood, and they had as much of an appetite for poetry as their predecessors did -- but with a far greater ability (especially among the poor) to purchase and consume it privately. It was a market that inspired strong competition between authors and publishers, and authors already known as major public figures for their poetical works became more celebrated yet. The production and consumption of poetry was something of a national pastime, and the war provided a much expanded venue for poetical work in styles and on subjects both new and old.
At the war's outset, something exciting had been happening in the world of English poetry -- the birth of what was called Georgian lyricism. The Georgian movement (if we may call it that) had been spearheaded by the influence of the editorial team of Edward Marsh and Harold Monro; it was comprised of a varied collection of young, dynamic poets producing modern works that were simultaneously deferential to many of standards of Victorian poetry and experimental in their subjects, moods and approaches. My somewhat glib summation of the Georgian experience is that it would be like seeing Walt Whitman and Robert Browning engage in a fist fight that ended with them making out. In any event, they were, I repeat, thoroughly modern -- but it is easy to forget this in light of the triumph of the Modernists, who have rather swept the field were that sort of name is concerned.
The reason the Georgian movement matters to what we're considering here is that it was home to a number of authors whose names are now far more familiar to us in connection with the war -- names like Rupert Brooke, Siegfried Sassoon, Isaac Rosenberg and Robert Graves. The works of these authors appeared in Marsh and Monro's Georgian Anthologies, which first started appearing in 1912; Brooke was in from the start, and Sassoon, Rosenberg and Graves began to be included with the third volume in 1917 -- as much for their longstanding friendships with this entire literary circle as for the power of their works. Some of the most famous of the war poets, then, do not necessarily emanate strictly from the trenches, but rather fit into a large and pre-existing literary movement that was already increasingly popular.
This doesn't necessarily explain the popularity of all of the soldier-poets, and certainly does not explain their proliferation during the course of the war. I'd wager that few unpublished souls would have intently scanned the lines of the Georgian Anthologies while waiting out a barrage near Ypres and thought, "that could be me." So what was it?
Competing Theories
A number of theories have been proposed, and I don't know that any one of them offers a total explanation for the huge presence of this sort of poet on the war's literary scene. Certainly there's the huge appetite for poetry already described, and the attendant curiosity on the reading public's part about poetical works being produced by those experiencing new and titanic events. But this public literacy was not a solitary affair; this was the first national army in British history that was itself broadly literate even (mostly) down to the lowest ranks, and this offered a much larger scope for written reflection on the soldier's part than had previously been likely in the wars of ages past.
Certainly the possibility you mention in your question is a compelling one as well -- life in the trenches, as is often the case in modern war, could be described as long stretches of boredom punctuated by moments of appalling terror and danger, and this provided many soldiers with the sort of time necessary for poetic reflection and refinement while simultaneously confront them with the sort of experiences and images that could inspire the kind of response that has now become so famous.
Pursuant to this, it is popularly and often said that the soldier-poetry of the war is born of an intense need to express the inexpressible -- to cross the gap that existed between the home front and the firing line in a way that allowed communication between the two worlds, and for those still at home to receive a vivid and "truthful" impression of what was being seen, experienced and felt at the fighting front. The degree to which this information was otherwise prevented from reaching civilians by censors, propagandists and the like has been somewhat exaggerated in the intervening years, but it remains the case that many of the war's poets felt themselves to be in reaction to the glib emptiness of newspaper prose or the appalling vulgarity of patriotic music-hall expression. "All a poet can do today is warn," wrote Wilfred Owen in his preface to an intended anthology of his poems -- it was never compiled, at least by him, as he was killed a few days before the war ended in November of 1918.
Whatever the cause of their popularity at the time, this question of perceived "honesty" has been a hugely important factor in the soldier-poets' persisting popularity. Ernest Hemingway, in an introduction to a collection of war writing in 1942, insisted that the work of the soldier-poets was essentially the only good and true work to come out of the war at all; "one reason for this," he continued, "is that poets are not arrested as quickly as prose writers would be if they wrote critically since the latter’s meaning, if they are good writers, is too uncomfortably clear." For Hemingway, the writer needed to be "of as great probity and honesty as a priest of God" -- while many of the war's soldier-poets were, many of the prose writers and patriotic poets of the war, in his estimation, were not. Whether this is actually the case (I would argue that it's a massive oversimplification), it's a perspective that remains common where popular comment on the poetry is concerned: there is propaganda on the one hand, and the unmediated experience of the war, as presented by the soldier-poets, on the other. It's what saw Virginia Woolf praise Sassoon's first volume of war poems in 1917 as being work which makes civilian readers say "'Yes, this is going on; and we are sitting here watching it,' with a new shock of surprise, with an uneasy desire to leave our place in the audience." It is work that inspires, but also convicts, in a way that so much of the war's patriotic home-front poetry now fails to do -- not, I would argue, because it was always incapable of so doing, but because we are now cut off from the concerns that motivated it in a way that we are not when it comes to the soldier-poets' dark lyrics of fear, pain, disappointment, and all-conquering irony.
Final Thoughts and a TL;DR
In closing, I'll note that the most popular anthology of soldier-poetry in English during the war was E.B. Osborn's The Muse in Arms (1917); it included the work of over a hundred poets, of whom almost all were on active service when they wrote their works. It sold well, and contains many names (like Graves and Sassoon) who have now become famous. In Osborn's preface, however, he offers a start warning that serves, even at that early stage, as a fitting summation of all that I have said above:
Of all the vast mass of civilian war-verse, very little indeed will survive; [. . .] it has nearly all been cast ere now into the waste-paper basket of oblivion. The making of verse memorials is perhaps the only task to which the non-combatant poet may address himself without fear of losing his sincerity...
This anxiety over sincerity has haunted the ongoing reception of the war's poetry ever since it first started to appear, and I don't see that ending anytime soon. It, as much as anything, accounts for some of the soldier-poets' popularity.
TL;DR: The war produced a lot of poetry full-stop, but the work of the soldier-poets is among the most popular that remains because of its lingering immediacy when compared to the patriotic homefront works that often focused on ideals and concerns that we find difficult now to share. The war's poetry was produced both in and by an atmosphere of greatly increased mass-literacy, and in a cultural atmosphere in which responding to major events with poetry was not only common but expected.