The most important thing,
@AcererakTriple6, I feel, is: what do you want their theme to be? Not their role in the party--we don't need to know if they're meant to buff or heal or fight or whatever right now. That comes later. Paladins are oath-sworn warriors. Warlocks gain knowledge from sources that humanoid dare not name. Druids are emissaries of nature.
Good question! This will of course differ from person of person, but I'm going more off of what the PHB describes as a Bard. Basically, the Magic-Storyteller (whether those stories be told through song, poem, or folktale). To me, bards are a very "traditional" class. If you're read any of Brandon Sanderson's Cosmere books,
Hoid is the epitome of my mental image of a bard (he's even called a "Worldsinger", as he travels the world(s), telling stories of other lands and folktales, and using magic as a means of telling the stories, occasionally playing musical instruments, as well as using the stories and music as a means to do magic). They (traditionally) travel the world, spreading stories and songs, and using magic to aid people around them (and to enhance their stories).
The magic comes from the stories. The weavings of fairy-tales, the stories that their old mentor taught them, and the words that they speak containing truths about the world.
So, simply, magical story tellers. The rest comes later, but that's the base. Songs tell stories, as do poems, but that's the source of their power. The knowledge of words of magic. 5e bards even have the "Words of Creation" story in their flavor-text that's expanded by the College of Creation. That magical words and stories created the world, and the bards have been using those to do magic. (This also relates to the magic of playing D&D, as the stories of our adventures and experiences together bring us closer together and feel like magic. IMO, bards would embody this. The fact that stories are why the world/game exists, and the channel that power and seek out more stories to enhance their power. The reason, IMO, bards level up is because as they experience more of the world and get more stories to tell, they're able to find out more about the magical Words of Creation and the purpose of stories in the world. At their core, bards are about emotion and magical stories, and using those to build/spread their own stories.)
Right now, bards are mostly a musical hodge-podge of a class who knows things. It's not a great theme, I know. But with that theme in mind, it actually kind of makes sense that they get such random spells.
Yeah, and that's kind of the source of the issue. Their theme is a hodge-podge, they're kind of the joke class in D&D. Besides the stereotypical "horny bard" archetype, bards are mostly known for stepping on the toes of a lot of classes' themes (Rogues as the major one, but also Enchantment/Illusion Wizards, Sorcerers, and a few others), which is where the issue lies. If their theme is largely "stepping on the toes of the other classes", that will be reflected in their mechanics (Expertise, Jack of All Trades, Magical Secrets, etc).
So: focus on their musical knowledge. Instead of making them half-casters, make them more like warlocks (but MAD, focusing on both Cha and Int). Few spell slots (but they get those slots back quickly) and a host of invocation-like abilities that are dependent on them playing music, telling stories, or reciting history or myth. Actually make them a knowledge-based class.
I disagree. I don't think that making them Pact-Casters would fix anything. They'd still have the issue of having non-thematic 6th-9th level spells (through their equivalent of Mystic Arcanum), have an even bigger issue figuring out what the heck they do in combat every round (as Warlocks primarily spam Eldritch Blast, but Pact-Magic Bards wouldn't even get that), and how you emulate "magical musicians/storytellers" through Invocation-like features.
I think that Half-Casters would work better, especially as I think that bards should rely on their subclass a bit more, and Warlocks are one of the subclasses that rely on their subclasses the least of any of them.