Unclear, though game designers seem (unofficially) ok with it
The list of things that Mage Hand can do is quite short, but the terms are quite broad. As quoted in your answer (and from PHB, p. 256):
You can use your action to control the hand. 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. You can move the hand up to 30 feet each time you use it.
The hand can’t attack, activate magical items, or carry more than 10 pounds.
It could be argued that you could use a Healer's Kit by "manipulating" it: after all, the first definition of "manipulate" via Oxford Languages (the dictionary service Google draws from at the time of this answer) is:
1.) handle or control (a tool, mechanism, etc.) typically in a skillful manner.
It certainly sounds like this would give you the ability to skillfully use the Healer's Kit with Mage Hand. But there are a few impediments to this idea.
Can you use the hand to do something that normally takes an Action?
Since you needed to "use your action to control the hand," it's reasonable to ask whether the "use" of the hand could include something that normally requires an Action: after all, you have no Actions left (barring the use of a feature that gives you a second one, like Action Surge). While this is an open question, we have some guidance in the list of prohibited uses, specifically the rule that (ibid):
The hand can't attack
The interesting thing about this prohibition is that attacking normally also takes an Action (typically, the Attack Action). So this text implies that there might be some forms of "manipulating an object" that (normally) cost an Action that mage hand can perform: otherwise, they wouldn't need to point out the Action-costing things that it can't do. This is known as an "exception that proves the rule."
It's not quite that clear cut, though. For example, the fact that some uses of mage hand explicitly allowed to an Arcane Trickster (e.g. "You can use thieves' tools to pick locks and disarm traps at range", PHB p. 98) could be done by "manipulating an object," but you clearly can't do them with regular Mage Hand since they are "additional tasks" (ibid) that an Arcane Trickster can do and other people casting Mage Hand cannot. So there are definitely Actions the hand cannot do which aren't listed in its spell description.
All in all, there is room to interpret the spell in either direction. But there is enough wiggle room that it is possible the hand could perform certain activities that normally would require a full Action to do, even though it already takes an Action to control the hand.
Does the Healer's Kit count as "an object"?
On the face of it, definitely not. The text states (PHB, p. 151):
Healer's Kit. This kit is a leather pouch containing bandages, salves, and splints.
So the kit is more than one object: it is a 3 pound collection of objects in a leather pouch. However, it's worth noting that when you are stabilizing a creature with the kit, we're not sure how many of these objects you actually use! Perhaps only one of the objects in the kit is needed for particular types of injuries (e.g. a salve for slashing damage, a splint for bludgeoning damage), or perhaps you need to use multiple ones for each type.
Now, medically, it would be entirely unrealistic to suggest that most treatments for serious wounds could be achieved with a single isolated object (I only have first aid training, but still this is easily apparent). Even bandages are actually several layered items, often gauze followed by a bandage to hold it on, sometimes preceded by salve. MivaScott points this out clearly and convincingly in their excellent answer. However, the application of actual medical logic is complicated here, because no realistic assessment of injuries is going to match with this game's rules for their treatment (e.g. no realistic injury system would have a person be entirely unimpaired in terms of their ability to perform tasks after one hour of unconsciousness, when they were previously inches from death).
Also, unlike any real world medical interventions, a Healer's Kit is (normally infallible at stabilizing a dying patient: it requires no skill Check to succeed, or proficiency to use, but simply always works (as long as the user expends one of its ten "uses"). Thus, we have real world medical materials (splints, salve, etc.) being applied in an unrealistic way: it's unclear exactly how this is done, and thus how many "objects" are involved. At best, we need to see this as another case of ambiguity.
So if the wording is all ambiguous, whom can we turn to? (The people who wrote the words)
In times of ambiguity, it can be useful to turn to designer commentary to clarify the rules. And there is designer commentary on this question, but it is the most unofficial and qualified kind. Specifically, the following exchange occurred on twitter with Jeremy Crawford (lead designer of the Player's Handbook):
Edward Krusling: Can Mage Hand use a Healer's kit to stabilize at range? If so, can Arcane Tricksters use the kit with a bonus action?
Jeremy Crawford: I'd allow (1) mage hand to use a healer's kit it's not holding & (2) an Arcane Trickster to do it w/ a bonus action.
Now, first of all, tweets by game designers are unofficial guidance in general, unless they are collected into the Sage Advice Compendium (which this is not). And second of all, by saying "I'd allow," Crawford is indicating what he would permit as a DM, which can go beyond rules interpretations (and via "rule 0" can even contradict the Rules as Written).1 Specifically, saying "I'd allow" means he is making a judgement call which is not demanded by the rules. That could mean that the rules are silent or unclear on the issue (which certainly seems to be the case to me), or that the rules require some other conclusion but he is overriding them as DM. In the absence of further commentary2, we have no way of knowing which way he meant the comment.
However, in the absence of other clear guidance, this exchange could be helpful. It shows that it may be possible to use the kit with Mage Hand, but there is enough ambiguity that such a decision is ultimately up to a DM.
And honestly, that's the best answer I can give to this question. The rules are sufficiently ambiguous that a DM will need to make their own call. But if, for whatever reason, you'd like to make the same call as the person who wrote most of the game (and I stress: you do not need to do that as a DM if you do not want to, and that decision was made by him as a DM, not as a game designer), then you should allow Mage Hand to use a Healer's kit to stabilize at range.
1.) It's also worth noting that the caveat written into his response (that mage hand can use a healer's kit "it's not holding") points to a possible high cost for this tactic in terms of the action economy. The text of the spell indicates it might take an action to open the kit ("open a... container"), another action to remove the object needed to treat the wound ("stow or retrieve an item from an open container") and a third action to actually use that item (" manipulate an object"). It also takes an action to even cast the spell, and the consensus (which I'd probably rule differently as a DM but is accurate via RAW) is that the Action taken to cast Mage Hand is separate from the Action required to control it. Thus, depending on your DM's assessment of how a Healer's Kit is "used," it could take three or four rounds for someone to "use" the kit with Mage Hand (which is a long time for a creature to be unstable and dying).
2.) Note that a twitter user did reply to the aforementioned tweets by Crawford, asking "Is this ruling on Mage hand and a Healer's Kit RAI or a house rule for your personal game?" Crawford has not replied to that follow-up question.