Your concept of what "myth" means is thoroughly modernist, and directly contradictory to what Tolkein meant. A bunch of damned liberals started using "myth" to mean "false rumor," and an antonym to "fact," and now that's what everyone thinks it means.
A myth is a story used to tell a deeper, archetypical truth. There are things we cannot explain, and so we write stories to illustrate a truth we cannot fathom. But whereas allegories teach us of specific instances, and demonstrate one-to-one representationalism, myths are universal.
The bible is a great example of something which is often both historically true and mythically true. When Christians teach us that we have crucified Christ, they speak in a mythical sense; I have not held physical nails to Jesus' flesh. (Yet it is also historically true that Jesus was crucified.) And whether it is historically true or not, we have all shared in the fault of Adam which gained for us a redeemer in Christ. While David wrote of his tribulations in war, we find in his writing the mythic truth of our struggle against evil; while Solomon wrote of his love for his Beloved, we find in his writing the nature of God's love for man.
Tolkein's goal was to tell us stories about the universal nature of man, which are true, even though they had not been revealed through history the way the bible was. Thus, he wrote myth. In comparison, allegory is much shallower; it is the mere attempt to present stories of specific temporal incidents or individuals in the language of symbolism and fancy.