Before we dive into the problem itself, let's talk about two little things that often get in the way of such structures: specialization and energy efficiency.
In nature really what matters is 2 things: not dying and reproducing successfully. Not dying requires that you have enough energy to survive, as well as the means to get that energy, but if you're a colossal mass of undifferentiated cells, chances are that you'll be outcompeded by another clump with specialized ones, because when it comes to energy efficiency, it's best to divide your "workers" so that each is responsible no more than a small couple of functions. If everyone must be responsible for everything, everyone is burning huge amounts of energy for no reason, and chances are that the whole process might actually becomes slower, because they're not able to focus on that one thing alone.
This strategy of specialization is in fact so successful that its present on nearly every single euchariotic cell in the world, them having special organelles that each handle a different function necessary for the cell's survival, and this specialization is also present in our brains, which is why our brain is divided into different regions that do different things (the part that handles vision is very different from the part that handles abstract thinking, which is different from the one coordinating the so-called "autonomous" processes of your body).
In reality, at no point does any part of our brain lies there waiting for its time to shine, they're all working all the time, never stopping even while you sleep. In fact, should a part of it stops working, it usually means you had a stroke or something similar, and that the part in question, or at least a portion of it, has died.
So in comes the question: why isn't a brain that can fully concentrate on one thing and one thing alone at a time be necessary better? Well, for you to be capable of doing pull-ups and play the piano like a master, even if you're never doing both simultaneously, you necessarily require both the strength needed to do pull-ups and the delicate finger movements/knowledge to play the piano, and given how usually delicate movements become less delicate as you become stronger, it means it's highly unlikely that you'll ever be as good at either as someone who can focus on only one of those. As the old saying goes: "jack of all trades, master of none". Being flexible with where you can get food and live is usually a good survival strategy, but it also makes you less competent than something fully specialized at doing that one thing or living in that one place.
Similarly, not only is it impossible for your brain to fully redirect itself to do only one thing (since, you know, your organs would all stop if it did that meaning you'd be at high risk of death), but even having only a couple of its structures redirected to be able to do that is highly likely to result in an overall higher average energy cost (which is a very bad thing when our brains, already very energy efficient, use up 30% of all the energy in your body, the cost of big-brain power being a selecting factor on why intelligence isn't necessarily always a good thing for every single creature).
Instead we already have a brain that, being specialized, can work on various things at the same time with little additional energy cost. Sure, being able to solve a math equation in your head in seconds is neat, but when it comes to survival, being able to dynamically use the various parts of your brain in unison to hunt prey, avoid becoming prey yourself is better. It's precisely because survival is usually reliant on efficiency, dynamism and flexibility that specialization is overall a more successful strategy over the more costly option of only partially specializing them.
As for your idea of moments of high priority to certain senses, it's called a flight or fight response, and it actually causes changes to your entire body,increasing breathing rate, suppressing functions such as digestion and immune response, "heightening" your senses and others. Similar physiological changes also happen in when your mammalian diving response kicks in.
Overall, there's a disappointingly high amount of things our bodies could do, but often doesn't unless it sees them as necessary, because stronger, faster and more agile bodies are also more costly to maintain.