I'm not playing "a warlock", I'm playing Khalil ibn Rasul, who is the sum of his abilities and personality. A class isn't a descriptor of a character, just an ability package. The spells granted from bard can be just as easily fluffed to come from my patron's teachings as those few additional ones gained from going through a bunch of warlock levels.
You missed the point by about, oh, five hundred miles. The sorcerer has a definitive feel for the class built into its mechanics. The wizard has a certain feel in its mechanics. The same with the bard, cleric, druid, ranger, paladin, you get the idea.
Those mechanics do shape the way the game plays and feels. If you have a class who's spells primarily involve curses, it will have an impact on the game from someone who's magic tends to revolve around illusions and enchantments.
You can design your character's story any way you want to your heart's content, but that doesn't change that the warlock
class does have a definitive flavor and feel of its own that you CANNOT get by multiclassing to bard and playing that after you get Eldritch Blast at 2. Yes, the bard gives you more spells a day. They do not get the warlock spell list, the invocations, the class abilities.
Like it or not, mechanics and story are intertwined together. They inform each other, and help shape each other. It affects the options your character has.
The simple truth of the matter is that your Khalil WILL be different if you play as a pure bard, a pure warlock, or a hybrid of the two. It might not affect the background and personality, but it will affect your actions as well as how others respond to said actions. Trying to suggest that all classes are interchangable, like you are, is doing a major disservice.
Maybe you are confused, thinking that bards and warlocks are simply "another variation on arcane trickster." The fey patron kind of are, since that's a classical fey thing in D&D, going back to elves and the Illusionist class. Same thing with the bard and their magic. But the warlock class, as a whole, is not. Fiend, GOO, Vestige, Sorcerer-King, all the different patrons feel differently in a way that's not simply "another class with different spell mechanics."