Yes, because that makes sense in this case
Can it drag?
The description of mage hand does not say it can drag things. Rather, it says:
You can use the hand to manipulate an object, open an unlocked door or container, stow or retrieve an item from an open container, or pour the contents out of a vial.
Here we have a clear list of the things the spell can do. The question is, is this an exhaustive list, such that the spell cannot do anything not on the list, or is it an exemplary list, such that the spell can do other things that are like those on the list, but not enumerated?
In evaluating how to interpret the spell, we have two competing principles. The first is Spells do (only) what they say they do.
Spells do only what they say they do
Mage hand says it can 'open an unlocked door'. If I apply the 'Spells do' principle, mage hand can open any door, of any size or weight, even if the door is barred and latched, so long as it is not locked. Mage hand says it can 'manipulate an object'. If I apply the 'Spells do' principle, mage hand can drop a grain of salt into a beaker, but it cannot drop a pinch of salt (because each grain would be a discrete object, and it does not say it can manipulate multiple objects). Mage hand says it can pour the contents out of a vial. If I apply the 'Spells do' principle, mage hand can empty a vial, but it cannot push, drag, or roll an empty vial across a table, nor lift one into the air - only manipulate it once aloft.
Later in the spell description, we are told:
The hand can't attack, activate magic items, or carry more than 10 pounds.
While the first list is a list of things the spell can do, the second is a list of things that it cannot do. If spells do only what they say they do, the second list is almost completely redundant. If the first list is an exhaustive delineation of what is possible with the spell, we do not need a second list, because the spell cannot do anything that is not on the first list. Notice that neither list says that the spell can carry objects of ten pounds are less. Rather, this is implied by saying it can't carry things of more than ten pounds, but this implication only works by assuming that the first list is exemplary, not exhaustive. A strict application of 'spells do only what they say they do' means that mage hand cannot push, drag, or roll objects - but neither can it lift or carry them, because the spell does not say it can.
In contrast to the principle of 'spells do (only) what they say they do', we have the principle of 'things work the way we expect them to, except as noted in the rules.'
Things work the way we expect them to
"In D&D, everyday things—walls, gravity, bread, laughter—work the way we expect them to, except for when the rules say otherwise." (JC tweet). Looked at this way, the first list, what mage hand can do, is giving us a list of possibilities, but it is suggesting that mage hand can do many other unspecified things, so long as they are of equivalent nature. The second list, the list of things the spell cannot do, are limitations on the first list - things the spell cannot do even if they are physically equivalent, because they interact in other important ways with the game rules. You might consider the first list to be physical limits on the narrative power of the spell, while the second list is balance limits on the game power of the spell. Under the principle of 'things work the way we expect them to', mage hand can be used to unlock a door with a key even though it is not delineated in the spell description, because the exemplary first list says mage hand can be used to manipulate an object. On the other hand, mage hand cannot be used to unlock a door with a chime of opening, because the second, restrictive list says it cannot be used to activate magic items. Mage hand specifically can be used to pour out a vial, so it can also be used to pour out a beaker of acid into a cauldron - but it cannot be used to pour out a beaker of acid onto a creature, because it specifically cannot be used to make an attack.
It should be obvious by now that I think the 'things work the way we expect them to' is more useful for interpreting mage hand than 'spells do (only) what they say they do'. Neither of these two principles is specifically described in the rules. Both are considered guidance, and both are supported by multiple rulings in different instances. I don't think every spell should be interpreted 'as we expect'. I suspect that I would be generally more inclined to interpret utility spells 'as we expect' and attack spells as 'spells do (only)'.
If it can carry, it can drag
Mage hand can carry up to ten pounds (but remember, if 'spells do only' is applied strictly, mage hand can't carry objects, just manipulate them). The principle of 'things work the way we expect them to (except as otherwise noted)' says that if the spell can carry up to ten pounds, it should be able to exert a force of about ten pounds on objects in the world - in fact, a bit more, since the spell needs to be able to lift the weight of the ten pound object against gravity.
My real world expectations are that any force capable of lifting and carrying ten pounds should also be able to drag, push, or roll ten pounds across most surfaces. Of course, surfaces and weights with high coefficients of friction will reduce the amount of weight that can be moved - at least based on my real world expectations. 'Unless otherwise noted' in this case means that I have to use whatever rules do exist for this interaction. The PHB rules on lifting and carrying say that a creature can push, drag, or lift twice its carrying capacity. If mage hand can carry 10 pounds, then it can push, drag, or lift 20 pounds, but in doing so to have its speed reduced to 5 feet per turn (compared to mage hand's spell description speed of 30 feet per turn). If the spell behaves the way I would expect a force to behave, mage hand could drag a weight of 10 pounds at 30 feet per turn, or drag a weight of 15 pounds at 5 feet per turn. Treating it as such a force allows me to adjudicate many more interactions with the world then simply those listed in the spell description.
In this particular case, I find it much more useful to rule based on the 'things behave as we expect' principle rather than 'spells do only what they say' principle. In this particular case, I think the 'things behave as we expect' principle produces more useful results.