I think the Bard outperforms the paladin in almost every category except that nifty Aura. The bard is a much more flexible class that empowers many character concepts, and can even do a lot of what the paladin does by poaching their spells.
Seriously, a Bard can have find greater steed, banishing smite, or destructive wave several levels before a Paladin can.
Or just take good spells like banishment or fear or fireball or animate objects etc instead.
The class is irrelevant there. It's half caster vs full caster while a person could use sorcerer instead of bard and still access higher level spells faster.
Bards have advantages but armor and damage are on the lower end of the scale. I do prefer bardic inspiration to aura of protection because it moves with characters and can be applied in more ways.
It really more that the paladin, warlock, and bard came into their archtypes in 3e. 4e showed how they look at at full power. So all 5e had to do is revert the 4e versions into a more traditional frame. Ranger and Sorcerer didn't get such solidification.
That was part of my open test feed back. "I want more 4e than 3e in my 5e bard."
4e wasn't my favorite edition but it certainly wasn't the train wreck some people want to portray it as.
The issue with ranger and sorcerer is more people didn't want 4e conversions.
Bards tend to not be able to fight on the front lines using Martial Weapons. Although, being a Sword Bard might mitigate the weaponry concern.
Swords bard abilities are attached to the attack action. That creates a conflict between using flourishes and casting spells.
Valor bards add shields and get to keep that AC improvement without burning inspiration or while casting spells. It's the better front line option.
Either is likely to go rapier and DEX, but valor gains all martial weapons proficiency to go a different build.
Bards can use weapons like anybody else. The only major gap is level 5, when the Paladin gets a second attack and 2nd level spells to smite with. Then level 6 rolls around.
Indirect damage is damage.
Bards can use weapons but they don't get abilities that improve using those weapons. The higher the level the bigger that weapon gap gets.
It's the same issue with cantrips where other casters get better damage cantrips and most gain abilities that improve cantrip damage.
Bards improve skill use and inspiration instead of weapons and spells. They are great at support but tanking hits and inflicting damage are not shine moments for them. They tend to take extra focus just to hit mediocre in those aspects at tbe cost of other areas.