I think the answer may be that the purpose of "formal" dealings with times is to explicitly remove the troublesome intuitive concepts which lead to great deals of disagreement regarding time. Consider that you say:
The human would be able to answer instantly and directly, having an innate sense of time.
You make this statement axiomatically. It's an assumption. You assume humans can do this. Many have assumed similar. The result tends to cause all sorts of issues. The famous issue of the grandfather paradox is one example. An attempt to apply human intuition to a rather nuanced problem typically results in all sorts of problems, hence why it is considered a paradox. A formal approach to time is typically used to break this up.
There are countless examples of where the human innate sense of time breaks down. Consider the famous Ship of Theseus.
The ship wherein Theseus and the youth of Athens returned from Crete
had thirty oars, and was preserved by the Athenians down even to the
time of Demetrius Phalereus, for they took away the old planks as they
decayed, putting in new and stronger timber in their places, in so
much that this ship became a standing example among the philosophers,
for the logical question of things that grow; one side holding that
the ship remained the same, and the other contending that it was not
The result of this was the classification of two extreme views on time, the endurable point of view which roughly models time as a series of frames like a movie, and the perdurable point of view which explores objects as things in "4 space," 3 dimensions plus time. We do not, in fact, confine ourselves to static thinking, unless we choose to limit ourselves to only using the endurable point of view.
The innate sense of time is also made far more complicated by scientific discoveries about how the mind work. It is becoming harder and harder to treat our perception as metaphysical hocus pocus; we are having to come to grips with the physical processes that appear to make the universe work. One key reality is that a surprisingly large amount of what we do can be modeled using local equilibrium thermodynamics, where the "relaxation rate" of small scale events is so much faster than the larger scale events we care about that we can handwave them away as error terms. Only a small portion of physics calls for non-equilibirum thermodynamics to achieve acceptably accurate predictive results. For systems where local equilibrium thermodynamics is sufficient, modeling time as a series of snapshots is actually a very effective approach. It is so effective, in fact, that in a surprisingly large number of cases people don't even realize they are making an assumption when they fall into the endurable point of view.
Of course, if you look at a large portion of the literature, the endurable approach is popular. Personally I believe that its because the questions that arise from the perdurable point of view are troublesome. In many cases, we want answers, so we turn to the more tractable endurable approach.
You will find a different focus in Eastern literature. Much of Eastern philosophy revolves around the Dao, which is much more reasonably modeled as a dynamic perdurable entity rather than an endurable series of frames. The Dao is ever changing, in their philosophies. As a result, they tend more towards perdurable ways of thinking. The result is that they require many concepts, such as Chi and the Dao, which seem strange to Westerners because they, with a typically endurable mindset, do not fully understand why the concepts are useful to the Eastern philosophers.