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.
The description of the Use an Object action says:
When an object requires your action for its use, you take the Use an Object action.
So any nonmagical item that specifically requires an action can be used with the Use an Object action. More specifically, everything on your list, with the exception of potions as you noted, can be used with the Use an Object action.
Best Answer
Objective Facts
You already found the writer-provided option for disarming - the Disarm option in the DMG. It is an Attack, not Use An Object; they're two different kinds of actions.
The uses of Fast Hands is fairly well laid out - locks, traps, and Use an Object.
The types of things Sleight of Hand is meant to apply to is fairly well defined:
Taking a weapon out of somebody's hands, a weapon they're actively using, is not legerdemain or trickery of any kind. The examples clearly deal with small objects - coin purses or something that fits in a pocket.
Subjective Opinion
I would not have ruled the way you did - partially because I was already aware of the DMG having a rule for generic Disarming (which I don't use), but moreso due to the Battlemaster Fighter having a Disarming Attack. Granting the Rogue easy (via a bonus action) access to something that a Fighter has to consume resources (Superiority Dice) to do is too much.
I would certainly allow the Rogue to use Fast Hands to relieve a character of a weapon they were not actively using by cutting straps and the like. Some may consider it a bit of a stretch, and I probably wouldn't let the Rogue actually take possession of it (just cause it to fall at the target's feet), but I think it's reasonably fair for a subclass-specific bonus-action ability to force another character to waste an action to pick the item back up (or make the choice to abandon it).