I realized my comments were better suited as their own answer.
You are absolutely correct that a hooded lantern would be visible in the dark from any distance. Assuming you had direct line of sight on the lantern.
So if you are in an open empty field, or a very large open cavern, then you are right...there's no point to a hooded lantern.
However, you can only see light if you have line of sight on it. And, when seeing 'light' there are two things you may be seeing. You may either be seeing the source of the light (the lantern) or you may be seeing the area illuminated by the light.
So, where this becomes useful is in areas that do not have clear, long-distance visibility. Such as in a building, or a cave, or a forest, or city streets, or literally anywhere that is not an open field.
The way this works is this: If you have a hooded lantern and you have the hood up, you are casting a Sphere 60' aura of light around you. In an area with corners, doors, or any other obstruction that blocks line of sight, this means any creature that can see any part of that 60' radius aura of light...can see your light.
However, if you hood the lantern, that drops it to throwing off a radius 5' aura of light. Bearing in mind that spheres include their origin as part of the radius...this means that only the single square that the lantern is sitting in is illuminated, with a bit of bleed-over into the neighboring squares. This means that unless a creature gets line of sight on the squares immediately around the one the lantern is in, they cannot see the light.
There are several practical uses for this. While dungeoneering, the party can dim the lanterns to sneak up to a corner. If each party member carries a lantern, they can each see where the other is, and see the ground under their feet so they don't trip or anything...but no one around that corner can see them coming, because none of the light created by the lantern reaches around the corner to become visible. Then the elf sticks their head around the corner, using their Darkvision to see what's there...all without ever showing any light to the creatures around that corner.
On the other hand, if you wanted to try the same trick with a non-hooded lantern, the party members that can't see in the dark would have to stay 60' away from the corner, so that the light created by their lanterns didn't go past it and reveal their presence.
So, in summary...
Yes, a light is visible from a tremendous distance, IF you have line of sight on the light source, or anything the light source is illuminating. In an open field, this means dimming your light doesn't help much. But in an area with obstructions, dimming your lantern means you reduce the aura that is visible in the dark from a 60' sphere, to a 5' sphere. And given that this aura can shine past corners, under doors, through windows, and so on....reducing your light aura is very useful in areas with obstructed vision.
I would rule that the original and the duplicates all have the effect of faerie fire visible. Faerie fire would outline all the duplicates and give advantage on attack rolls regardless. However while the spell does say
A duplicate can be destroyed only by an attack that hits it. It ignores all other damage and effects.
Mirror Image also says
Until the spell ends, the duplicates move with you and mimic your actions, shifting position so it's impossible to track which image is real.
I think that the duplicates would not be affected by faerie fire directly BUT seeing as the original is affected, they would alter their appearance to match that of the original thus conferring advantage to attacks made on them as the advantage is a result of faerie fire's outline making it easy to see the affected creature. (Otherwise why would the spell have to specify "if the attacker can see it"?)
Best Answer
You must supply oil
In general, the idea of putting a Light-infused object into a hooded lantern in order to benefit from the radius-reduction effect is probably sound, and clever. But if "spells only do what they say they do," then we should apply that to magic items as well.
The item's description clearly states
Emphasis mine. So they must be in the lantern's light, not the light of a magical rock carried by the lantern. And how does the lantern make light?
So the item clearly states that the magical effects come from the light that the lantern creates when burning oil.