I am going to try to walk through this problem, in a step-by-step manner in relation to exercise, starting from at rest, and ending at the point in which the body is no longer able to maintain its energy-charge.
The body mainly utilises oxidative phosphorylation to maintain its energy-demands. In cells where great amounts of energy need to be produced very quickly, eg. cells that are actively replicating, glycolysis is the preferred mode of energy production because this pathway is able to very quickly produce large amounts of ATP, and lactate is able to quickly diffuse in to the bloodstream. Lactate will travel in the bloodstream, to the liver, to be recycled back to glucose through an anabolic process called gluconeogenesis. This recycling pathway is known as the Cori-cycle. Essentially, what is happening is, parts of the body that are in need of high amounts of energy, will "dump" their wastes to the bloodstream, to be dealt with by other organs.
Science and Skiing VI, 2015, pp 17-30. George Brooks of the University of California summarises lactate recycling quite nicely here.
Muscle cells will begin to quickly utilise ATP-stores, and release glucose from the glycogen-stores, and release oxygen-stores from myoglobin.
As these stores begin to become depleted, the body will begin to go in to overdrive, in an attempt to restore itself to a resting-state. Thus, your heart-rate will increase, breathing will become faster, and glycolytic pathways will be activated through feed-forward mechanisms.
I have touched on this before, in another post that you may find interesting. I believe that if you read through this though, that you will understand why there is low levels of oxygen, and what the body does to attempt to circumvent this.
There will be very low levels of oxygen throughout the active tissues. The body will do everything it can to try to restore this, but it will ultimately fail to do so. Thus, the only real option it has is to rely on glycolysis in the tissues that are causing "problems". The rate of ATP-production is most dependent on the rate at which a cell can take glucose in to the cell.
The electron-transport-chain is only able to synthesise ATP at its maximum velocity - this maximum velocity is proportional to the concentration of oxygen. The rate of glycolysis is, however, essentially, only affected by the rate at which lactate can be removed from the cell and the rate at which glucose can be absorbed by the cell.
Of course, the muscles that are being used will only be concerned with producing energy. Therefore, all of the lactate that they produce will be dumped in to the bloodstream, where it will travel to the liver, in the previously mentioned cori cycle.
Overall, the issue is not in the muscle-cell's ability to regenerate NAD+. This is the easy part, seeing as there is no net-change in the concentration of NAD+ when glucose is converted to lactic acid. And also, things such as malate-aspartate shuttles, citrate-pyruvate shuttles, and glycerol 3-Phosphate shuttles can be used to maintain redox-balance. These are only but a few examples of how to maintain redox-balance.
You must also remember that carbohydrates are not the only class of molecules involved in these cycles; degraded proteins and fats can also be used to supply the TCA-cycle with alternative means of energy-production. Pyruvate probably would not even be able to be converted in to acetyl-CoA anyway, since pyruvate dehydrogenase requires the presence of oxygen to function.
Problems however do arise once the body is unable to deal with all of the waste that it is producing. There will reach a point, where the liver cells will be physically incapable of accepting any new lactate-molecules. And so, lactate will accumulate in the bloodstream. With an accumulation of lactate in the bloodstream, lactate will be unable to diffuse to the outside of the cell. Thus, cells will be unable to convert pyruvate in to lactate.
The reason why ATP-regeneration eventually fails to keep up with the rate of ATP-usage, is because the rate of conversion to lactate decreases. The rate of conversion to lactate decreases, because the concentration of lactate increases
I tried to make a little illustration of what I mean above, below. Hope this helps to conceptualize it.
In this second picture, there is too much lactate for diffusion to occur.
Taken from Quantifying intracellular rates of glycolytic and oxidative ATP production and consumption using extracellular flux measurements; Mookerjee et al. 2017
"These maximum values of JATPglyc and JATPox define the bioenergetic capacity of the cells. As shown in Fig. 5D, the maximum individual capacities of JATPglyc and JATPox in the bioenergetic space plot intersect at (62.5, 46.5) for a theoretical maximum bioenergetic capacity of 62.5 + 46.5 = 109.0 pmol of ATP/min/μg of protein. At this maximum point, the glycolytic index (GImax capacity) would be 62.5/109 = 57.3%, making C2C12 myoblasts primarily glycolytic when running at their maximum ATP production rate. Compared with the actual value of JATP production in the presence of glucose (55.2), the bioenergetic capacity was 109/55.2 = 197% of the rate with glucose (Fig. 5D). This bioenergetic capacity of 197% of the rate with glucose (alternatively, a reserve capacity of 109.0 − 55.2 = 53.8 pmol of ATP/min/μg of protein) reveals that the C2C12 cells under our experimental assay conditions with added glucose were operating comfortably within their capacity to generate ATP and were well set up to respond to any acute increases in ATP demand by increasing either glycolytic or oxidative ATP production, or both."
In these experiments, the researchers were trying to figure out what proportion of ATP would be created by glycolytic and oxidative pathways in muscle cells (and other cell-types). Their findings, were that, indeed, the majority of ATP production in muscle cells comes from glycolysis.
During exercise, this difference in amounts of ATP production between glycolysis and oxidative phosphorylation would probably be even greater.