Mechanically speaking you left out very important mechanics, rules about dueling ability, weapon properties, creature size, and interaction of all those. Plus the comparison should be between a Lance and a Pike, the closest comparison.
Lance - IMO the two handed property was not added to allow the classic jousting scenario with a shield. Clearly the lance requires 2 hands to use unless you have some other assistance, this is reflected in being mounted.
Since this was not added small creatures can use it use it one handed.
Since anyone can use it one handed you can use dueling. Dueling was added to some martial classes to reflect their greater skill in using smaller weapons, this evens out the damage trade vs a larger weapon for those classes. I don’t think it was intended for the Pike, the lances closest relative, see rules, and thus not a pike used mounted.
The dual wielder feat was intended to use 2 full size weapons at once, not 2 lances from a mount. I don’t KNOW this but I suspect that was the case.
I get that it is RAW, but this isn’t a balance issue, it’s an absurdly issue. It’s a fantasy game, but I don’t think I know a DM that would allow a halfling on Golden retriever with 2 lances attacking with each and then get a bonus action attack again.
Or another way, how would players react if a goblin rode up on a wolf and attacked 3 times with a lance. “Wait a minute here!” Would be the mildest response.
But if your DM does a small Paladin on his mount should always use 2 full size lances and dual wielder feat combined with 2 weapon fighting and then just switch to two small weapons if need be.
Ok so, i listed the weapon properties in the pro and con - some not by name but by effect because saying "special" did not seem to be very informative.
Second, you do not get to pick which and only which weapons get to be compared. i chose two for their melee strength and sane damage dice to avoid having to try and assess the value of pike d10 vs lance d12.
But if you want to add it to the mix fine
lance
pro - 1d12 damage + reach
con - two handed unless nounted, disadvantage within 5' (tyhe most common melee range)
greataxe
pro 1d12 damage
con - heavy and two handed.
pIKE
PRO REACH
CON d10 HEAVY TWO HANDED
Lets assume pike is the norm - the baseline.
Great-axe adds extra damage and removes reach. A gain and a drop. Are they equal? Varies by campaign.
lance added extra damage, changed the disadvantage penalty feature (heavy) from "for small creatures only" to "for everybody at 5' range only" and reduced the two-handed to "mounted only." So, ups and downs and mixed bag of nuts to be sure but the "mounted only 1 handed" thing definitely carries some heavy baggage - that whole horse or other mount type thing.
So, again, not seeing the built-in weapon imbalance there.
Now, i can only guess that by "dueling ability" you may mean the Dual Wielder feat and not the Dueling fighting style or some other "ability" you may be thinking of... like maybe a sub-class ability? i do not know.
But regardless of what you were referring to - if it is a feat or a class feature then the issue is not with the lance at all but with whatever class feature changes that lance.
Its like when people want to yell and fuss about how powerful eldritch blast is and compare it to other cantrips... but want to include three or four invocations and warlock hexes into the mix *for one side* and not take into account other class anf feats when comparing.
For example, after you make sure and claim what all i left out...
lance and dual wielder feat...
compare to greataxe and great weapon master feat for the 5/10 and bonus action "cleave" thingy
compare to pike (or others) with pole arm master for the bonus attack and Ao on entry.
I do not see this as screaming OMG the lance is broken to me when it gets the feat in play.
As for what you feel, what you prefer, what you divine as the reason behind the shadow of the intent of the dream of what someone else thought lance properties were to represent... thats between you and your mirror.
as others have pointed out, martial skill could be defined and depicted as every bit a magical effect as anything else can. So maybe its the bound macguffins holding up the lances for the paladin.
they style of a campaign is determined at the table and very fantastic style and tone games might well be just fine with the war dog mounted pally halfling with his lances.
you want a little more down to earth campaign, great! All power to you. Simply try and put in place rules you all agree on to reflect that and ways to discuss out the edge cases that go beyond RAW and printed rules.
neither is right or wrong.
neither deserves scorn.