The culture that spawned the concept of the Bard is almost unrecognizable to a modern viewer and requires a signficant shift in how you think about music. The modern American culture is one of the least musical cultures that has ever existed, not in that we don't like to listen to music, but we find it odd to participate in making music. Singing and dancing is something that performers do and is not something that is considered particularly masculine. It's certainly not something we think leaders do. We think leaders listen in silence and are entertained by music. D&D does itself no favors in portraying the bard typically as a sort of roguish dandy, rather than as the mystical warrior priest and battle leader that is the origin of the concept of the D&D bard.
To get back to that concept, you have to embrace the Northern European notion of music as being intimately tied to martial prowess that is almost extinct except in the bagpipes of Scotland.
"Théoden seized a great horn ... and he blew such a blast upon it that it burst asunder. And straightway all the horns in the host were lifted up in music, and the blowing of the horns of Rohan in that hour was like a storm upon the plain and a thunder in the mountains...
Ride now, ride now! Ride to Gondor
the battle-fury of his fathers ran like new fire in his veins, and he was borne up on Snowmane like a god of old, even as Oromë the Great in the battle of the Valar when the world was young. His golden shield was uncovered, and lo! it shone like an image of the Sun, and the grass flamed into green about the white feet of his steed. For morning came, morning and a wind from the sea; and the darkness was removed, and the hosts of Mordor wailed, and terror took them, and they fled, and died, and the hoofs of wrath rode over them.
...And then all the host of Rohan burst into song, and they sang as they slew, for the joy of battle was on them, and the sound of their singing that was fair and terrible came even to the City."
Theoden embraces his role as battle captain by making music. This is Theoden displaying Bardic power. Note how Tolkien sets apart and italicizes what Thedoen king says. He sounds a note on his horn, and he doesn't merely shout out the words, he poeticly sings out: "Ride now, ride now! Ride to Gondor!" And what is the result? Theoden himself is filled with battle magic, and a rage that makes him like he was young again, and he becomes a visige of righteous power and and the light of good. And the whole host takes up his theme, and - probably lead by the battle captains - begins to sing a slaying song, a war song, to the death of their enemies.
There is nothing at all ridiculous or corny in this scene. You just have to accept that music itself isn't ridiculous and corny. You have to believe as the Gauls of old believed, that it is entirely proper and powerful to sing during battle.
Another example of the origins of the Bard is in the Kalevala, where the heroes make magic by singing. When another character in the story submits, they do so by confessing that the other is a stronger singer than they are. Väinämöinen notably knows a song that causes his foe to sink bodily into the ground. In this origin, which was also very influential on Tolkien, we see other echoes in The Lord of the Rings in the character of Tom Bombadil.
Tom arrives to help the Hobbits anouncing to the world his skill at singing:
Old Tom Bombadil is a merry fellow,
Bright blue his jacket is, and his boots are yellow.
None has ever caught him yet, for Tom, he is the master:
His songs are stronger songs, and his feet are faster.
He then sings a song of banishment to prove it:
Get out, you old wight! Vanish in the sunlight!
Shrivel like the cold mist, like the winds go wailing,
Out into the barren lands far beyond the mountains!
Come never here again! Leave your barrow empty!
Lost and forgotten be, darker than the darkness,
Where gates stand for ever shut, till the world is mended.
The big problem with playing a Bard is that to pull it off well so that you don't come off as corny, you actually have to be able to improvise like that. Otherwise, no one is going to be able to imagine what you are doing that isn't corny.
