The Invisible condition expressly states that it causes Disadvantage on Attackers trying to attack them, but in doing so, doesn't adjacently make reference to whether the creature in question can see them:

An invisible creature is impossible to see without the aid of magic or a special sense. For the purpose of hiding, the creature is heavily obscured. The creature's location can be detected by any noise it makes or any tracks it leaves.

Attack rolls against the creature have disadvantage, and the creature's attack rolls have advantage.

Note that this is distinct from the section concerning whether Invisible creatures can be seen.

Meanwhile, Truesight obviously permits the ability to see an invisible creature:

A creature with truesight can, out to a specific range, see in normal and magical darkness, see invisible creatures and objects, automatically detect visual illusions and succeed on saving throws against them, and perceive the original form of a shapechanger or a creature that is transformed by magic. Furthermore, the creature can see into the Ethereal Plane.

... But it doesn't actually say anything about negating the Disadvantage/Advantage components of the Invisible condition.

What I am forced to conclude is that, while a creature with Truesight is perfectly capable of seeing a creature that is invisible, they would still suffer disadvantage when attacking them, and their opponent would still have advantage against them.

The converse situation is something like Faerie Fire, which specifically handles Invisible creatures with special language:

Any attack roll against an affected creature or object has advantage if the attacker can see it, and the affected creature or object can't benefit from being invisible.

Just as a reference for explaining why I suspect this doesn't apply to Truesight.

Now, to be clear: I consider this to be an asinine ruling. At any table I would DM, I'd be ruling that Truesight does, in fact, negate the effects of the Invisible condition, since this seems more like an oversight/mistake than an intended reading of the rules. But is there something I'm missing, that causes my RAW reading to be incorrect? And if this is the intended outcome, is there a good reason for me to accept it as-is, and not overrule it for my tables?

Unseen Targets and Attackers:

When you Attack a target that you can’t see, you have disadvantage on the Attack roll. This is true whether you’re guessing the target’s location or you’re targeting a creature you can hear but not see...When a creature can’t see you, you have advantage on Attack rolls against it. (PHB p.194)

Thus, the advantage/disadvantage mentioned for invisible creatures comes from being unseen, which is negated by True Sight. Since the character with True Sight can see the invisible creature, it no longer has the "invisible condition" in respect to the character with True Sight, but for everyone else it still does,

The example spell you gave (Faerie Fire) is a different case. Faerie Fire cancels the benefits of being invisible, but does not cancel the invisibility.IE. You see the faerie fire outlining the creature, but do not see the creature. And since everyone can see the faerie fire, everyone knows where it is.

Summary:

The two spells are worded differently, because Faerie Fire cancels the benifits of being invisible for everyone that can see the fearie fire, while True Sight only cancels the invisible condition for the character that has True Sight.

From discussing this issue in chat (starting here), the conclusion I've come to is that my RAW interpretation of these rules relies on accepting literally the PHB definition of the Invisible condition, which is both superfluous and poorly written.

It should have confined its effects to stating that the affected creature is treated as "Heavily Obscured", which itself sufficiently justifies the "Advantage/Disadvantage" effects of the condition, and a proper reading of the condition should presume this case instead of relying on the strange decision to separate out the "Advantage/Disadvantage" effect from the "heavily obscured" effect.

Part of the reason this seems correct is that, if we were to make my original RAW ruling the correct way to parse out the effect, it would make spells like See Invisibility or True Seeing (and by extension, the Truesight feature) much weaker than they should be, rendering them nearly useless in some cases, like if the creature is too noisy to properly Hide its location in the first place. Especially a spell like True Seeing, being a level 6 spell, it would be very strange if it barely negated the effect of the Invisible condition.