Though I know that RAW states that Vicious Mockery is initiated through a "string of insults", my question is about what would happen if you used it elsewhere than a tavern brawl or the like.

As a reference, the cantrip Vicious Mockery is written as:

You unleash a string of insults laced with subtle enchantments at a creature you can see within range. If the target can hear you
(though it need not understand you), it must succeed on a Wisdom
saving throw or take 1d4 psychic damage and have disadvantage on
the next attack roll it makes before the end of its next turn.

This spell's damage increases by 1d4 when you reach 5th level (2d4),
11th level (3d4), and 17th level (4d4).

Unlike similar opinion-effecting spells like Friends and Charm Person, the wording of this spell doesn't finish with a caveat "when the spell ends, they will know you influenced them and it will piss them off", revealing that, it's the insults and not the subtle enchantments that will upset them.

Given this, my understanding of how the cantrip works is that you insult someone, using magical enchantments to boost the persuasiveness of your words. If they fail a wisdom saving throw, they take your words to heart, shading their next move with a veil of sadness, doubt, or fear clouding their judgment. As your words sting, they take minute psychic damage.

If that's the case then, I'm curious to know what would happen if your character tried lacing the same subtle enchantments while attempting other feats of persuasion?

Would your words of endearment seem more authentic when using it to court a potential partner? Would your opinion seem more valid when using it to settle an argument? Would the opinion someone has of you become more likable when trying to dispel doubt in an argument? ... Or will everything sound like an insult?

Though I realize that this does not guarantee the character's desired outcome and the added effects can be negated if they succeed a wisdom saving throw, I'm wondering if it would be a way for a character to increase their chance for a favorable outcome without the other person feeling swindled or used in the end...

No, Vicious Mockery is not a charm

While vicious mockery is a spell from the school of Enchantment, none of its capabilities include charm effects or the charmed condition.

The spell itself is not an opinion-effecting spell - it is a damage and debuff spell. You can see this in the tags on dndbeyond and note that it does not carry the social tag.

Mechanically, the spell only does what it says, which is:

You unleash a string of insults laced with subtle enchantments at a creature you can see within range. If the target can hear you (though it need not understand you), it must succeed on a Wisdom saving throw or take 1d4 psychic damage and have disadvantage on the next attack roll it makes before the end of its next turn.

These effects are not subtle. You have cast a spell, dealt psychic damage, and done something to make it more difficult for them to succeed on their next attack.

A possible way forward

This doesn't mean that a DM could let you use it in this way, but just that the general mechanics for it don't include it. If you did use this, and the target didn't fully engage in combat back, you may be able to pair it with a more social skill of intimidation, but that is going to be entirely up to the DM and wouldn't be that difference than threatening someone in general after hitting them with a melee or ranged attack.

Narrating effects

It's also important to consider the narration process. This is a highly personal piece, but I generally approach by narrating to support the mechanical effects. Narrating as a means of explaining may introduce more confusion than necessary and create misunderstandings of what a spell can do. Some caster classes, like the Bard, are limited in what spells they know. Stretching spell capabilities beyond that minimizes that limitation - but if that matters is highly table dependent and would be a creative use of a spell.

Not in the way you want it to.

Vicious Mockery isn't just "influence them to be a bit demoralized". If you continue to viciously mock someone, you can literally kill them. It's not so much making your words extra-persuasive as it is using those words as a channel to assault their mind. Worth noting that Vicious Mockery outright requires line of sight. If you can't see the target, you can't affect them with the spell. At the same time, it doesn't require that they be able to understand you - just that they be able to hear you.

Further, and more specifically, in 5e, spells do what they say they do. This one tells you what it does. It inflicts psychic damage. It doesn't do things like persuading or flattering. That's not doing psychic damage, and therefore not what the spell does.

However, at least technically, yes, you can use Vicious Mockery to win arguments or gain favors. If you meet some ruffian accosting a helpless NPC, and engage them in impassioned debate, then you could use Vicious Mockery to deal psychic damage as you inform them of details of their parentage about which they were previously unaware. If they should happen to yield the floor to you and flee (perhaps because they did not wish to die from psychic damage) then you will have won that argument by default... and the poor innocent NPC in question (ears burning) might well feel that they owed you a favor or two.