[RPG] the advantage of Ranged Legerdemain for Sleight of Hand

No, the first two points don't hold for any caster with Mage Hand.

Pickpocketing isn't just taking something from a pocket; it's a subtle skill which requires doing so unnoticed, after all. This is more than just "doing it while the victim's back is turned" - the human body registers slight touches and subtle sensations, like the weight of an object. A pickpocket knows how to beat these senses - by touching the victim in other places to confuse the senses, for example.

But the Mage Hand is implied to be clumsy, with limited dexterity - not conducive to stealth.

(Note that nothing in the Mage Hand description says that the hand has significant dexterity - it can "open an unlocked door", for example, or "pour the contents out of a vial", but apparently not manipulate lockpicks or disarm traps, which requires more precision. It's limited almost exactly to simple tasks that you don't have to practice.)

So it's not that the Mage Hand eliminates the normal skill check as such - instead, the rules as written do not contain a "normal" skill check for picking a pocket with a Mage Hand. (A standard thief can't pick locks with a Mage Hand either, even if they somehow have one.)

Most people can undo the buckle on a bag, or shoe. But can you do so so swiftly and subtly that the holder of the bag (or wearer of the shoe) doesn't even notice? Try it. It's not as easy as it sounds. It needs significantly more manual dexterity than just being able to undo a buckle - it's more like being able to play a piano.

Under the old-school skill system of 3e or 4e, the correct way to handle this would have been with a skill roll and an extremely hard DC - as GM, personally I'd have set the DC 10 or even 20 points higher than the usual for that kind of pickpocketing. The old skill system would then have allowed top-level characters to pull it off anyway.

But 5e discourages this "everything is technically possible with the right roll" approach, in favour of limiting skills to "actions anyone could attempt". "Pick a pocket with a magical force" isn't something anyone could attempt, and there's no obvious RAW reason why being able to pick pockets by the normal means would help you use a Mage Hand to do so.

(And it is typically next to impossible to pickpocket successfully with a fully visible, somewhat clumsy, disembodied hand. So even a disadvantage is not enough penalty - it should just be impossible.)

Legerdemain clearly gives the hand more dexterity, not just invisibility. This allows for more complex tasks.

Short answer

If the standard hand doesn't have enough manual dexterity to pick locks or disarm traps, it doesn't have enough for the equally tricky task of picking pockets.

Yes, it's a pretty obvious secret door.

As you say, passive Wisdom (Perception) is 10 + WIS, so anyone with a non-negative WIS modifier will passively notice it.

But that doesn't mean you definitely see it.

The interesting exception I see is if your party is rushing through the area. Per Travel Pace moving 400'/minute or 4mi/hour will earn you a -5 to passive perception (PHB p.182). Lots of characters don't have the +5 WIS modifier necessary to offset that, and might blow right past it.

@Dale M points out the more likely scenario when this may happen: disadvantaged passive checks. By rule (PHB p.175), a passive score receives a -5 bonus when subject to disadvantage.¹ Low light, obscuring conditions, or other environmental challenges could create disadvantage, similarly challenging one's ability to notice a DC10 secret door.

¹ The "disadvantage=-5" rule of thumb is a lamentable addition to 5e culture. But, as this excellent answer shows, in the specific case of DC10 it's the perfect modifier. The second graph in the linked answer -- the vertical distance between the black and blue curves, to be precise -- shows this to be the case.