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.
Bracers, no. Shield, yes.
The description states, as you have pointed out:
They surround the wearer with an invisible but tangible field of force...
(emphasis: bold) The bracers themselves emanate this invisible tangible field of force and would have to be the target of any other special effect.
Magic Vestment, for a target, states:
Target: Armor or shield touched
There is no "suit of armor" - just a "force field."
The shield, however, would be fine as a target.
Note: An animated shield would already have an enhancement bonus1 (since its cost is a +2 bonus), and your spell would merely over-right, or if your caster level isn't high enough, or have no effect on the enhancement bonus the shield already has.
1Magic Armor: A suit of armor with a special ability must have at least a +1
enhancement bonus.
Best Answer
A creature can take something from a mage hand but there are no explicit rules for this situation
The spell text for mage hand doesn't state the conditions under which an item could be taken from the spectral hands grasp.
However, the game does provide DMs with tools, which I have applied as follows in this situation in my own game.
Isn't the mage hand pretty weak?
The rules state that '[t]he hand can’t carry more than 10 pounds.' But this doesn't explicitly refer to it's grip strength and so doesn't necessarily mean that, once it's holding an object, that an object can easily be taken from it by a much stronger PC. It might be a reasonable logical step - but real world physics or logic don't have any official bearing on the rules for this situation.
It's up to your DM to decide how much weighting they'd want to give this reasoning. They could decide that the mage hand's perceived weakness makes removing an item from its grasp essentially trivial, if so no roll would be required. Otherwise...
...this could be a resolved via a contested check.
From your wording 'The DM had the other PC make a Strength check against my Strength score' it's not clear to me if this was a contested check - it sounds like your DM used your strength score to set a DC, rather than having you both roll, as you would in a normal contested check.
In this situation, your character's strength has nothing to do with the strength of your mage hand. So a contested check should probably more appropriately pit the opposing PC's strength against your spellcasting ability (intelligence for Arcane Tricksters).
If your DM wants to account for the apparent weakness of the mage hand in this situation (as discussed above), as part of a contested check, then I would advise employing the rules for advantage and disadvantage here:
So, in summary, in my own games I have ruled that this situation should be resolved via a contested strength check from the opposing PC against an intelligence check with disadvantage from the Arcane Trickster.
As an aside, don't forget that 'you can move the hand up to 30 feet each time you use it', so to avoid items being stolen from your mage hand in the future, you could simply keep it floating high up enough so as to be out of reach of most others, whenever it was not needed.