Only beginner weightlifters can gain muscle and lose fat at the same time." For intermediate and advanced weightlifters, they can only cut or bulk, but never both at the same time.
You can do whatever you want... you won't necessarily be successful with it, or you may not see optimal results, but that doesn't mean there is only One True Approach and nothing else can work.
Defining the "paradox"
To build muscle, your body needs a caloric surplus to give it energy. In contrast, to lose fat your body needs a caloric deficit so it will convert fat into energy. Losing fat and gaining muscle, therefore, appear to be opposing goals. Therefore, a beginner losing fat and gaining muscle would be a paradox.
How does a beginner lose fat and gain muscle at the same time?
Well, first off it should be noted that there are not very many studies done on this, so you should challenge the expectation that a beginner can lose fat while gaining muscle. Perhaps their measurements were not methodical and accurate, or were obscured by outside variables? Maybe they had a base amount of muscle that was hidden by fat, and losing fat only made it appear like they were gaining muscle?
Assuming that the paradox is true, you must revisit your assumptions as they were either incorrect, incomplete, or both. The first assumption that is made is the time period in which the caloric deficit/surplus occurs must be a day. A 24 hour period is a convenient unit of measure for counting caloric intake, but perhaps not for our body. It is possible that with a healthy and well-timed post-workout meal your body is able to build muscle while still running at a caloric deficit to lose fat.
Another assumption that must be challenged is that both outcomes have to be positive at the end of the day. Well that is not the case because we aren't interested in losing fat and gaining muscle for one day, but rather for the long term (say, over the course of one month). So even though you may have gain 0.4g of muscle and gained 0.2g of fat on one day, the next you may lose 0.1g of muscle and lose 0.3g of fat. So while neither day saw the "paradox" of both gaining muscle and losing fat occurring at the same time, the long-term effect was a net positive in both directions.
Why does this biological process shut off after the beginner phase?
In your beginner phase you have the most explosive growth, both in lifts and in muscle; however this is not a linear growth, as there is a definite upper limit in which your growth tapers off as you come closer to your genetic potential. This brings us back to what I was saying earlier about the "optimal" approach. As a beginner you have such explosive growth that you get away with the less-than-optimal, but as an advanced lifter you are so close to your upper limit that you have to finely tune your program, meaning you have to go through the bulking and cutting cycles.