We are told that photosynthesis, developed a couple of billion years ago has produced all the atmospheric oxygen. I wonder, why did the process stop? Why do modern plants prefer to recover the entropy whereas ancient ones...
Actually, the ancient ones separated the carbon-hydrogen fuel from the oxygen. The modern trees consist of this fuel which re-joins with the oxygen afterlife. But why didn't this happen to the ancient plants? Where did the ancient carbohydrates go? If it is the oil and gas we can burn it. Why do you say that we do not have enough fossils to burn all the oxygen? Where are the ancient plant bodies deposited so securely that we say for sure that they cannot come back into the contact with the oxygen?
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If I understand it right, you are assuming that in the beginning we had CO2, which was then split to organic carbon and O2 via photosynthesis. And now you are asking if it's possible to reverse all of that by burning all organic carbon, so that it consumes the O2.
Your assumption is not quite right. Oxygen is the most abundant element in Earth's crust and mantle, locked up in rocks. Rocks are made out of mostly oxygen. Atmospheric oxygen is negligible, when looked up in the context of the entire Earth. Oxygen is constantly being consumed and produced by reactions between the atmosphere and the solid Earth.
Not only that, there are processes in the Earth that take sediments and pull it down deep into the Earth, in subduction zones. Anything can be subducted: both organic unoxidised carbon and oxidised carbon in the form of carbonates (e.g. limestone). This goes down to tens to several hundreds of kilometres, far away from any interaction with the atmosphere.
As an example, let's say that you have a forest which that consumes a certain amount of CO2 and H2O, generating organic carbon-hydrogen and releasing the excess O2 to the atmosphere. Now you have carbon (and hydrogen) locked up in solids. Some of this organic material burns, decomposes, or otherwise reacts with atmospheric O2 to generate CO2 and H2O all over again. However, some of it is buried deep in the Earth's crust (that's how we get coal, gas and oil). So in theory, we can extract and mine everything, and then burn it. However, some of it is buried even deeper and is carried down bysubduction to the Earth's mantle. In there, it is far away from our reach and there is no way we can extract it to burn it. So the oxygen generated while producing this organic material lingers in the atmosphere, but the organic material itself is deep below in the mantle, unable to react with any oxygen (as O2, any redox reactions are a different story).
To sum it up, your point of view is too simplistic as it does not take into account the Earth as a whole, but only top soil and atmosphere.
Photosynthesis has not stopped. It happens all the time, splitting water and carbon dioxide, and producing oxygen and carbohydrates. Likewise, organic matter rots and decomposes all the time, requiring oxygen and releasing carbon dioxide. The Keeling curve actually illustrates this quite nicely: most of the land mass where this happens lies in the north, and during the summer photosynthesis dominates whereas in the winter, decomposition dominates. As a result, CO2 concentration in the atmosphere decreases in the northern summer and increases in the northern winter.
Your question about using up all of the oxygen in the atmosphere by burning all fossil fuels is a good one. If indeed you could get hold of all of the carbon that, over the past 2 billion years, has been separated from the oxygen by photosynthesis, then you could do that. But the carbon is not accessible: only a small fraction is actually in the form of fossil fuels. A much larger part is simply dispersed in the form of organic matter in the sediments of the earth (both on land and in the sea), and it may not be concentrated enough to actually burn by itself. For example, soil has a significant fraction of carbon, but it does not usually burn. More carbon has been subducted in sediments through plate tectonics and now resides dozens or hundreds of kilometers down in the Earth mantle. (A fraction of it will eventually come back out as carbon dioxide in volcanic eruptions).
In other words, most of the carbon that has resulted from photosynthesis is no longer accessible for burning, and consequently not available to react with the atmospheric oxygen.