I am a contemporary of both Dylan and The Beatles. I've never seen them as competitors. Yes, Dylan is, to me, a writer of more significance in relation to world and life realities, inequities, spirituality, and injustice.
However, 'more accessible', I think not for most. Indeed, I have known many who've rejected his songs on hearing his voice. It's a voice that can be grating. However Dylan uses it in a variety of ways and it can be as tender and soothing as it can be gruff and hard-hitting. One wouldn't know unless having heard and given thought to the extensive variety of Dylan songs. and singing.
The awards given to these artists are, in my mind, relatively irrelevant, as most awards in any field tend to be. In the case of Dylan and The Beatles, they are of different contexts. Personally, I would prefer that The Beatles had all continued to refuse 'Royal' or elite institutional honours, as they did when first offered. However, they are not primarily for music but for service to the nation, not least bringing in foreign currency!
The reality is, however, that Dylan and The Beatles have had quite different routes to their musical development and, to a large degree, I feel have appealed to somewhat different audiences and markets. Each has brought masterful, innovative and creative writing, arrangements and playing to their art. Dylan for me, is an unsurpassed lyricist because I have long been an activist for social justice, human rights and have tended to study and be aware of injustice around the world.
At the same time, as do most of us, I enjoy being warmed, feeling good, having relief from pain and The Beatles have often given me that. I remember first hearing Love me do - and really enjoying its merry simplicity. Revolution and my playing it too much in one particular pub once got me ejected and I still enjoy listening to its boldness. Let it be and Yesterday have to be two of the most moving lyrics, and perhaps I'm wrong to include it but I find Mull of Kintyre simply beautiful and sung brilliantly by Paul.
So, although I'd agree that there are Dylan songs of much more profundity, such as Black Cross, Masters of War, When the Ship Comes In, Let me die in my footsteps and the much more recent Tempest the reality for me is that Dylan is more poet and story teller and The Beatles entertainers. Both have their place and we have been fortunate to have all of their talents.
There are massive differences between these artists and a much more considered and accurate critical assessment could be made as to their abilities, innovation, talent or appeal than that which you've countered, which is trivial and mostly baseless, in my opinion, not least because it serves no useful purpose, sounding to me simply 'sour grapes'.
I have both artists in my collection and recognise a variety of quality in the work of each, with brilliance and its opposite sometimes on show. I am glad to have grown with both and sad that the Beatles, as a whole haven't survived but grateful that I live in an age when I can still enjoy their original sounds as well as the last chapters of McCartney, Staff and Dylan.