The most likely reason that fermentation stopped after just one week is the yeast simply ran out of nutrients/oxygen
Agreed except for this point. Fermentation ultimately boils down to redox chemistry. Fermentation is an alternative form of using sugars for energy that specifically doesn't require oxygen as a terminal electron receptor. Yeast most efficiently metabolize sugars, energy dense molecules made of carbon, oxygen, and hydrogen, in an oxygenated environment as every bond is broken down and has its energy extracted, leaving only CO2 and H2O as end products. Most of this energy is converted to the cell's "energy currency" ATP in the electron transport chain which takes the reducing power of chemical intermediates, and passes these electrons along to ultimately be added to oxygen (for every reduction, you need an oxidizer and oxygen is a good one.)
In fermentation, central metabolism is stopped due to this lack of oxygen. It has too many electrons and nowhere to put them. For non-fermenting beings like us humans, this is suffocation- your cells will produce no energy and will die off. Yeasts and many bacteria, however, have anoxic forms of energy production collectively put under the label of fermentation. These are a bypass of much of central metabolism (The citric acid cycle and the electron transport chain) in favor of using a relatively large chunk of the original sugar as the terminal electron receptor; this most commonly is lactic acid or in the case of yeast ethanol, that alcohol that humans love.
Fermentation is a bad idea, energetically speaking, whenever there's oxygen available. A complete use of a glucose molecule from glycolysis, the citric acid cycle, and the electron transport chain can is a net gain of 36 ATP/glucose. Fermentation to lactic acid or ethanol results in a net gain of 2 ATP/glucose. You don't have to be a math wizard to understand why organisms would prefer aerobic conditions vs anaerboic fermentation. There's also the fact that both ethanol and lactic acid is toxic if enough of them accumulate. For facultative anaerobes, organisms that can live both in aerobic and anaerobic conditions, the aerobic conditions have a lot going for them; though there's other reasons why oxygen can be bad for living things, the energy differential is massive.
So if you're making prison wine like cobes is and you're looking to maximize your alcohol content (taste be damned, because we are talking about Cobes here) at least from a biochemical level you want to be perfecting your sugar to yeast ratios while allowing the least amount of oxygen in. Obviously you have to do the burping because every aqueous solution is going to have some dissolved oxygen and you don't want your plastic jug to blow up from accumulating pressure. There's 2 main threats your yeast in there that will eventually kill it when you're brewing- either starvation because there wasn't enough sugar or environmental toxicity from the ethanol levels. Any oxygen added is going to help convert sugar into unwanted CO2 and H2O.