No the rules do not require you to "wield" or "draw" an Object to Use it with a Use Object Action.
I'm going to try and answer my own question based on, and summarizing, the excellent answers here and elsewhere. As the answers here talk about rulings and involve what might be considered houserules (as well as clear citations of the rules as written) but the original question was about rules or rules clarifications/erratas.
- There is no mention of "wielding" or "drawing" objects when taking
the Use Object Action in the rules.
- There's an actual example in the rules of using an object while having it stowed on your person: "withdraw a potion from your backpack". In this example from the rules the backpack is worn and is neither drawn, wielded nor in the characters hands but is used.
- Removing the backpack example from the rules is a houserule some (many?) people use and without it there's no other instance of such an example in the object interaction rules. Without it there is still no written requirement to wield or draw the item but people houseruleing away the backpack often seem house rule in this requirement.
The above ruling/houserule is done in the name of common sense and having objects in the game world behave as they would in real life. But it feels that what people are talking about here is: NARRATIVE SENSE. And it seems that this example is often houseruled away solely because the timing of it makes bad narrative sense, rather than because the backpack is not in the players hands when used.
The spirit of the rules do, and your GM should require that actions make narrative sense (see Groody's answer). So narratively during the Use of an Object the character will be required to use their hands, as such the character must have one or two available hands (depending on object) during the Use of the Object otherwise the action fails to make narrative sense. What's not required by a written rule is drawing or wielding that Object into the characters hands using another action.
One thing that I found while researching this was a tweet from J. Crawford that stated:
An improvised weapon is, indeed, a weapon, but only the moment it's
used as such. A chair/shield/etc isn't a weapon otherwise.
I'm not going to pretend I now exactly how he intended this be resolved as he's made some rulings I feel make for very unsatisfactory narrative experiences (which I wont enumerate or debate here). As such I do understand that he and I are not always on the same page about rulings, and he's the authority and I'm just a person on the internet. That said here's my resolution of that having given the matter some thought based a very educational discussion with Groody:
- Character: Uses an oil flask from their pouch, knapsack, etc as part
of a Use Object Action.
- DM: checks rules: No mention of an object needing to be in hand to use. Though this DM does state that if the item was stored somewhere less accessible, like a backpack, they would step in and make a ruling that using the flask from the backpack would require an additional action. They cite the example of "withdraw a potion from your backpack" from the rules as the upper limit of what can be done with a Use Object Action at their table.
- Character: makes an attack roll as per the oil flask usage entry.
- Narratively: the flask enters the players hands from their belt and is thrown (Only with more dramatic wording and flare).
- Oil flask becomes a weapon
- Oil flask hits target, breaks, and is no longer a weapon
In this procedure the Oil flask has been used while it is an object so adheres to the object rules.
As Groody points out it's incorrect to assume objects function like ammunition if there is no wording saying they do. And by that sound judgment it also seems incorrect to assume objects like ropes, backpacks, etc function like weapons if there is also no wording to that effect (There is in fact wording showing you don't need them in hand, see: "withdraw a potion from your backpack" example).
Objects that change classification during use are more complicated I guess. But I don't feel like the above take on the matter is a bad one as the classification changes when used (past tense), not before use.
As everyone has stated: the rules on this, as written, are vague and do call for DM rulings. But after I've thought about and discussed this with the very learned people here, it's apparent that there's no written requirement to have an object in hand to use it, though the DMG does imply that narratively there is a requirement to have enough hands (or appropriate appendage) available that the use of that object makes narrative sense.