Yes, this build is competitive. Compared with a strength based dueling paladin, you'll actually be better off in many ways. Though you can't compete with a great weapon fighting paladin in damage. To illustrate this: let's compare some general build options.
Let's assume there are two paladins: Strong and Dexter. Strong and Dexter have the same stats except that Dexter has dexterity equal to Strong's strength, and strength equal to Strong's dexterity. This means that for the purposes of attack rolls and damage modifiers, the two are exactly the same. For that reason, I'm omitting modifiers when I make damage assessments below. I'm also not including magic items because with enough magic, you can make anything work.
Sword and Board
Dexterity is very competitive here, if not the best choice. In this case, both paladins take the dueling style and wield a weapon in one hand and a shield in the other. The highest damage one handed weapons all deal 1d8 damage. For Dexter, that means using a rapier. For Strong, it could be a longsword, warhammer, battleaxe, or rapier. Regardless, because the damage die is the same, and the paladin's relevant ability score modifiers are the same, their damage output is identical. Eventually, when Strong gets plate armor, they will have 1 higher AC than Dexter does (18+2 vs 17+2, as you have noted) Strong also has options to deal all three types of physical damage (slashing, bludgeoning, and piercing). Dexter will only be able to deal piercing damage, but their initiative, stealth, and dexterity saves will all be higher than Strong's. Dexter also has much better ranged attack options since they're as effective with a bow as with their rapier.
In return for being 5% easier to hit compared to Strong with plate armor, Dexter will be better at range, go earlier in the initiative more often, and make their dexterity saving throws more often than Strong does. Honestly, that's a pretty fair trade off.
Great Weapon Fighting
Dexter can't compete with Strong's damage here. Dexter keeps their shield and rapier. But Strong takes great weapon fighting and a heavy weapon. Their damage die goes up to either 1d12 or 2d6, depending on which weapon they choose. And they get to reroll 1s and 2s on that damage. Using the results of the How much damage does Great Weapon Fighting add on average question, that works out to roughly an extra +1 damage on average. So, strong will be dealing about 8 damage per hit before modifiers regardless of which weapon they choose. Strong could also take the Great Weapon Master (GWM) feat to further increase their damage output. Dexter is still dealing about 4.5 per hit before modifiers with their rapier and doesn't really have any feats that will consistently increase their damage to Strong's level. So, Strong is dealing twice as much damage as Dexter on average. However, Strong loses their shield. So even with plate armor, Dexter now has a 1 AC advantage (17+2 vs 18), making them 5% harder to hit than Strong. And Dexter is still better at initiative, ranged attacks, and dexterity saving throws.
Here's an interesting option. If Dexter has 13 strength, they can take one level of fighter to pick up the two-weapon-fighting style. They also take the Dual wielder feat so they can use two rapiers. Now, Dexter and Strong have the same AC (17+1 vs 18) and Dexter deals an extra 4.5 damage on average with their off hand attack. up until 4th level, Strong will deal about 8 damage per hit and Dexter will deal about 9 damage over two hits. So Dexter hits harder than Strong! As long as both attacks hit anyway.
The issue is that once the paladins hit 5th level they get an extra attack. Strong now deals 16 damage over two hits (8 damage twice), and Dexter deals 13.5 damage over three hits (4.5 damage thrice). And that doesn't factor in the bonus damage that GWM offers Strong. At this point, Strong hits harder than Dexter does, and again Dexter can't do much to catch up.
This is a very quick analysis. Enough to get the point across I think. There is no way for a Paladin using dexterity to deal the same damage as a paladin that's focusing on dealing as much weapon damage as possible. If you want to deal maximum damage, you have to go with a strength paladin. But dexterity gives you much better survivability. Dexterity saving throws are very common, and can deal some really high damage (think fireballs, lightning bolts, and fiery dragon breath). And with higher dexterity you can go earlier in the initiative and position yourself where you need to be as soon as possible. Plus the large damage bursts that paladins are known for because of their smite is, as you have pointed out, not dependent on weapon damage. It's just based on spell slot level. All of that is to say, the survivability and flexibility is something that will definitely serve you well and I personally think is a reasonable trade-off.
I wondered this before myself, I found that Wikipedia had most of my answers.
The Forgotten Realms is a fantasy world setting, described as a world
of strange lands, dangerous creatures, and mighty deities, where magic
and supernatural phenomena are quite real. The premise is that, long
ago, the Earth and the world of the Forgotten Realms were more closely
connected. As time passed, the inhabitants of planet Earth have mostly
forgotten about the existence of that other world – hence the name
The AD&D 2nd edition Player’s Guide to the Forgotten Realms Campaign (TSR2142) explains on page 2 the name is based on its relation to our own world losing contact with and forgetting it:
What are the Forgotten Realms?
Some theorists explain dragons and other fantastic occurrences by postulating parallel worlds. In times past, they believe, travel between mundane worlds — like our own — and more exotic locales was easy and frequent. But we have lost contact, and lost the worlds themselves. Lost, or forgotten.
Abeir-Toril, more commonly Toril, is an Earth-sized planet dominated by a continent in its northern hemisphere. Called Faerûn in the west, it is here that the Forgotten Realms lie. […]
The holy avenger was officially part of the Forgotten Realms since the beginning.
The official beginning of the Realms in a game publishing sense1 starts in 1987 with the Forgotten Realms Campaign Set—the “Grey Box”—published for first edition AD&D. By that point the holy avenger was already an integral part of AD&D, appearing in the DMG (1979) on page 165.
Note that Supplement 1: Greyhawk is a 1975 supplement for the earlier, original D&D game (dubbed “original D&D” or “0e” (zero-e, oh-e)) published in a box of three stapled booklets in 1974. The holy avenger by any name was a well established part of the game before AD&D or the Forgotten Realms were even conceived of as publishing products at TSR.
The 1987 edition of the Realms predates the idea of AD&D edition changes and the accompanying cataclysms in the Realms. Those were later inventions by minds at TSR after Gygax had been forced out. This makes the holy avenger part of the baseline Realms from which every edition change and cataclysm later derived.
So for the purpose of tracing the origin of the type of magic sword, it’s integral to the Realms, and its AD&D description is the baseline version, before any Realms events that changed how magic works.