Do bards really know what they're doing? Do they even 100% know they're 'doing magic?' I mean, being around them is magical, so...
Bards don't really 'cast' spells or 'use' magic. Magic is just something that happens around them- sometimes in response to their art, sometimes because the whims of fate smiles upon them.
FIRST-LEVEL BARD SPELLS
Animal Friendship: Self-explanatory: The animal is just friends with you now!
Bane - works as written, some reality-warping plus demoralizing
Charm Person - being around a bard is kinda like this anyway. Might accidentally cast this on someone you're just trying to talk to.
Comprehend Languages through gestures, expressions, etc, you are able to communicate pretty well with any creature.
Cure Wounds - Awww, see? Just a scratch! You'll live to fight another day old chap!
Detect Magic - 'I got a feeling something weird is going on' - plus instead of school of magic, they'll know genre of story- horror, fairy, religious, epic, etc etc
Disguise Self - poorly-improvised disguise works unrealistically well.
Faerie Fire - I dunno, actual faeries show up? Maybe scrap it.
Feather Fall: You fall at normal speed, but something miraculously breaks your fall- water, a pile of hay, etc. Somehow, you take NO damage, even if the thing breaking your fall should only, at best, cushion it. On falls over a hundred feet, make a Con save to avoid being knocked unconscious. Maybe you land on a ledge?
Healing Word - 'you got this!'
Heroism - you describe to someone how cool they are, and they believe you
Hideous Laughter - you tell a particularly hilarious joke
Identify - probably a lucky guess, plus remembering something you read in a book one time
Illusory Script - You're just super good at writing in code and innuendo
Longstrider - probably scrap this just 'cuz it's so boring
Silent Image - ugh. I WANT bards to have illusion magic. Maybe they just know a little bit of actual, intentional magic? Or maybe it works by people misinterpreting things that are already there- a bundle of sticks in the corner becomes a horrifying monster, etc. Maybe the bard makes a supernaturally realistic painting out of improvised materials.
Sleep - Can't be cast in, but basically works retroactively- for example, guards have a better-than-average chance of being asleep ALREADY when you try to sneak by.
Speak with Animals - He says he likes me better :)
Thunderwave - You shred a riff SO AGGRO it sends opponents staggering backwards clutching their ears
Unseen Servant - also kinda boring
To be clear, this IS all magic- it can be dispelled, countered, etc. It's not even that hard to notice necessarily, especially if it KEEPS HAPPENING. It just doesn't really originate from the bard, like a wizard storing spells in their skull, it more like originates AROUND the bard- like a big puppy following them around. At higher levels, the magic becomes more overtly... magic. And the cross-class spells the Lore bard gets (in 5e) are just that- wizard spells, sorcerer spells, etc, and are cast as such.