One of the the liquids than an alchemy jug can produce is 1/2 ounce of basic poison. However, there is no indication in the item's description of how much you can do with this amount of poison. In the description for a poison vial, it says you can coat 1 weapon or 3 pieces of ammunition with the poison from one vial, but it doesn't say how much poison comes in a vial. So, is the half ounce of poison from an alchemy jug intended to be equivalent to one vial? If not, how many weapons or pieces of ammunition can be coated using the half ounce of poison?
There are no numbers given for how much poison is contained in a vial of basic poison, how big that vial is, or how much a single "dose" of poison is so it's hard to say with 100% certainty and ultimately comes down to DM ruling.
A vial from the equipment list can hold up to 4 ounces of liquid (PHB, pg. 153).
Assuming a vial of basic poison contains a full 4 ounces of poison, then this gives us 25% weapon coverage per ounce of poison (or 12.5% per half-ounce), and 75% coverage on 1 piece of ammunition per ounce (or 37.5% cover on 1 piece of ammunition per half-ounce).
In short, you wouldn't be able to cover even one piece of ammunition, let alone one weapon, with the half-ounce of poison the alchemy jug produces and it would take you at least 8 days to produce enough poison to fill a vial.
However, page 139 of the DMG states that:
Most potions consist of one ounce of liquid.
Basic poison isn't a magic item or a potion but it is clear that a 4 ounce vial isn't the standard liquid container.
If a vial of basic poison is only one ounce then you could cover half a weapon or 1.5 pieces of ammunition and would only take 2 days to gain enough poison for full coverage.
Option 3 is to keep it simple. Don't get bogged down with unnecessary levels of finicky detail trying to figure out how many ounces are in a single dose of poison; the designers sure didn't in this instance.
If we keep things simple then the half-ounce of poison produced by an alchemy jug is equivalent to and follows the same rules as the vial of basic poison.
I think your first instinct is the correct one: the half-ounce of poison is a single dose, suitable for the standard one weapon or three pieces of ammunition.
How many ounces are there in a vial of poison? While "a vial" is an intentionally vague amount that could range from several ounces to just a bit of fluid, we can use weight to put some limits on it.
In the Player's Handbook, there are a small number of items with listed weight under a pound, which is of course 16 ounces. The smallest is the piton, at 1/4 pound, or 4 ounces.
Since a vial of poison has negligible weight, at most it must be somewhat less than 4 ounces in total -- and that includes the glass, stopper, and fluid together.
I don't think it's possible to narrow it down much more than that from weight.
But, logically, a vial of poison really can't contain multiple ounces of fluid, or it would be able to coat much more than three arrowheads (or one blade). Even accounting for waste, half an ounce seems like a reasonable amount of viscous fluid to smear on a handheld object without dumping it all over the place or having a lot of poison left over.
You are making this way too complicated. A basic poison does a minimal effect, most creatures are resistant or immune to it.
You can use the poison in this vial to coat one slashing or piercing weapon or up to three pieces of ammunition. Applying the poison takes an action. A creature hit by the poisoned weapon or ammunition must make a DC 10 Constitution saving throw or take 1d4 poison damage. Once applied, the poison retains potency for 1 minute before drying
The players can harvest this once per day. Let them build up an army of the stuff, if they take their action to apply the poison in the middle of a battle, then that is one less attack against a monster. It will be balanced out or potentially hurt them.