Why? I can cut as I like and write proper mathematical formula in new arbitrary coordinates. The model doesn't deny such a possibility even if I don't see any practical reason to do it.
No, you can't. You don't understand the allegory and you are too hasty. It probably partly comes from the fact that contemporary school program in physics doesn't give the classical Newtonian perception in its original way, as it was described in his "Philosophiæ Naturalis Principia Mathematica" (Mathematical Principles of Natural Philosophy). You can arbitrary choose the direction of spacious axes and the beginning reference point, but you are not free to choose the direction of the time axis. And the simultaneous events always stay simultaneous in any Newtonian reference system, because Newtonian Time is absolute. Newton has even attached a reference system to his Absolute Time and Absolute space. Regarding my allegory it means that you can cut the bar only at right angle. But contemporary pupils are not given this concept, they are told from the very beginning that any inertial reference system is equally good, but it isn't a Newtonian concept.
The case of cutting the space-time bar at angles other than the right one is also not arbitrary, the angle depends on the velocity of my reference system relative to other inertial systems, so you are not free to "cut as you like". But if you move you must cut at angles other than right. Then, if you imagine that a trace in a bar was made not by a tiny point, but a ball with a perceptible diameter, you'll find that the section of this trace may not be round but oval. It means that moving objects change their sizes relative to inertial reference systems other than their own. And the more the difference in velocity of the systems is, the more is the change in the size of the objects. This can never happen in Newtonian absolute space and time. Another allegory is that if we have traces of two events in the space-time bar, which can be described as two wholes in a piece of cheese they may appear on the same plane of one section and in another section there can be only one of them. So, as the sections along the time axis represent the moments of time it means that two events simultaneous in one reference system are not simultaneous in another and it also doesn't make sense from Newtonian point of view, but it makes sense from the relativistic point of view, and once again it shows that you are not free to "cut as you like" the classical Absolute Time and Space.
Natural sciences are objective - even if a scientist use formulations that may reflect his personality the essence is independent on that. Humans don't "invent" natural laws - they discover.
The history of science disproves this assertion. A very demonstrative example is the change in physicists' perception of the nature of light. Huygens believed that light is oscillation of ether, Newton reckoned that it's a stream of particles, corpuscles. Both theories had their strong and weak sides and in different times each of these theories experienced flowering and decay.
The truth is we study reality not directly as it is, but designing descriptive mental models that can fall near or far from the truth. The more we study Nature the more complicated our models became, the nearer to the truth we get, but I rather doubt if the definitive description of Nature is possible let alone there is no doubt we are only at the very beginning of the process of cognition.