Along with their French origins, Tolkien objected to the overt Christian
references. Considering Tolkien's own devout Christianity, I find that
fascinating. But for him, overt references to religion spoiled the myth.
And by the way, the original verson of the Quest for the Holy Grail was written
by French monks, as a way of indoctrinating young initiates into their
order. It is chock full of doctrinal references.
Tolkien does not make much mention of Celtic myths. For whatever reason,
fair or not, he does not seem to have considered Celtic myths sufficiently
English, as opposed to British. This despite the fact that he knew Welsh,
and was undoubtedly familiar with Welsh tales. He was an Anglo-Saxon
scholar, though, not a Welsh scholar.
Of course he knew about Beowulf. I don't know enough about Beowulf to say
why he rejected it as an English myth. Perhaps it is not long enough, or
is too Scandinavian. I'm not even sure if it takes place in England,
although it may have been written there.
When Tolkien said he wanted to create a myth for England, he meant The
Silmarillion, not LotR. And I am not sure he thought he had succeeded,
for he never published it. But LotR adds another element, the very 19th
century hobbits. As such it is not as mythic as The Silmarillion, but is
far more accessible to today's reader than any myth. If it were not for
LotR, I doubt that anyone would read The Silmarillion, let alone make it a best
seller.
So is LotR a good myth? I dunno, but the mythic elements sure make it a
great story! Surely there is room in this world for a new myth, and a new
story, as well as the old
ones.
“I dislike Allegory - the conscious and intentional allegory - yet any attempt to explain the purport of myth or fairytale must use allegorical language. (And, of course, the more 'life' a story has the more readily will it be susceptible of allegorical interpretations: while the better a deliberate allegory is made the more nearly will it be acceptable just as a story.)” (From Tolkien Letter # 131.)