Their hair isn't necessarily actually fire.
Looking back to earlier editions of the game, D&D 3e's Forgotten Realms Campaign Setting describes the fire genasi's hair as flame-like, but not necessarily made of actual fire:
They appear mostly human, with one or two unusual traits reflecting their quasi-elemental nature, such as skin the color of burnt coal, red hair that waves like flames, or eyes that glow when the genasi is angry.
The art in that book shows them with hair that resembles fire, but again, it may simply be that they have hair which looks somewhat like fire.
Indeed, the quote you give merely says that genasi with actual fire for hair are quite rare. Such a character would logically shed light like a torch, which isn't a standard feature of the genasi as statted, suggesting that most player character genasi do not actually have flaming hair.
However, such a creature would need to use fire-resistant material to build their pillow. Historically, according to the Wikipedia entry on pillows, the ancient Chinese developed ceramic pillows, while ancient Egypt may have used stone pillows. Many ancient peoples simply used their arm as a pillow.
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.
Best Answer
Yes, you can blind something with Truesight.
The only thing Truesight does in this case is magnify the ability that was lost to blindness:
Truesight enhances one's ability to see: through magical darkness, invisibility, illusions, shapechanges, and into the Ethereal (PHB p.185).
But the blindness condition now says "you can't see. The sight through illusions, darkness, &c. which you so enjoyed just a moment ago, you don't have anymore (PHB p.290, paraphrased)."
In your specific example of the Demilich, note that blindness is not one of the (many) condition immunities listed in its stat block (MM p.48).