I think the real problem is frequency.
We can all imagine really impressive and epic moments that involve breaking out into a haunting song or delivering an inspiring speech. But when your character keeps doing it, over and over again, you run out of neat ideas or those epic moments lose their impact and just because repetitive... or downright silly.
The easiest solution is just not doing it. Cast like a normal person most of the time, and only break out the big songs or inspirational speeches during the boss battles. That keeps you thematic enough that you still feel like a bard when it really matters, without watering the impact of your performances down with constant repetition.
Other than the fact that our cultural sensitivities are not attuned to the culture that produced the idea of the bard, and that we are ourselves no longer a choral culture that sings communally, I think that this is the real issue.
If I want to RP the greatest swordsman in the world, I don't actually have to be even a passable swordsman in real life. I don't even actually have to know anything about fencing, HEMA, or kendo - much less actually be able to perform them. The physical qualities of the character are completely distinct from my own physical qualities, because while the physical body of the character exists in the game space, my own physical body does not (and cannot).
The same is not true of social or mental qualities. You can't fully distinguish between the intelligence and charisma of the character and the player. You can mechanically advantage those attributes, and try to force them to be referenced during process resolution, but you can't (and wouldn't want to) draw a hard line between the character's mental abilities and the player's mental abilities. Lots of players feel that's unfair (and will even argue that it is wrong), but fundamentally both the mind of the character and the mind of the player exist in the mental space because the character is the puppet or avatar of the player.
One of the many implications of this is that you can't play the world's greatest Bard without at least be something of a great entertainer yourself. Even if mechanically your character can do all the things the world's greatest Bard can do, unless you yourself are capable of being entertaining, you can't actually RP that character. You yourself have to be charismatic, dashing, able to spout poetry, give stirring speeches, or maybe even break out into song, or else you're characterization will not be believable, will be awkward, and instead of inspiring will actually come off as silly. That's because you aren't actually the world's greatest bard, and if you were, it perhaps wouldn't come off as silly.