No, the first two points are not valid for any thrower with Mage Hand.
Pickpocketing is not just about taking something from a pocket. it's a subtle skill that requires doing it unremarked, after all. It's more than just "doing it while the victim's back is turned" – the human body records light touches and subtle sensations, such as the weight of an object. A pickpocket knows how to beat those senses – by touching the victim in other places to baffle the senses, for example.
But the Hand Mage is implied to be clumsy, with limited dexterity – not conducive to stealth.
(Note that nothing in Mage Hand's description says that the hand has significant dexterity – it may "open an unlocked door", for example, or "pour the contents of a vial", but apparently, do not not manipulate lockpicks or defuse traps, which requires more precision, it is limited almost exactly to simple tasks that you do not have to perform.)
So, it's not that the mage's hand eliminates the normal skill test as such; the rules as they are written do not contain a "normal" skill test to choose a pocket. with a mage's hand. (A standard thief can not take the locks with a mage's hand, even if he has one.)
Most people can undo the buckle of a bag or a shoe. But can you do it so quickly and subtly that the holder of the bag (or the wearer of the shoe) does not even notice it? Try it It's not as easy as it sounds. It is necessary much more manual dexterity as the simple ability to undo a loop – it's more like being able to play the piano.
Under the 3rd or 4th old school skill system, the proper way to handle this would have been with a skill roll and a huge disadvantage – as GM, personally, I would have awarded a penalty ranging from – 10 to -20. The old system of skills would then have allowed the high level characters to draw anyway.
But 5th discourages this approach "everything is technically possible with the right roll", in favor of limiting the skills to "actions that anybody could try". "Choosing a pocket with a magic force" is not something anyone could attempt, and there is no obvious reason RAW whether to be able to select pockets by the usual means would help you use a Mage's hand to do it.
(And it's usually almost impossible to capture with a perfectly visible, awkward, disembodied hand, so even a disadvantage is not a sufficient penalty, it should simply be impossible.)
Legerdemain clearly gives the hand more dexterity, not just invisibility. This allows more complex tasks.
If the standard hand does not have enough manual dexterity to take the locks or disarm the traps, it does not have enough for the equally delicate task of picking up the pockets.