You sure are having battles here, Vladimir! I find myself, however, having moderate sympathy with this particular question.
I think your comment, “I just do not see any physical motivation in it. I am afraid it is done by analogy with QED and that's it.” can be seen as a large part of its own answer. When we say that something is "Physically Motivated", I take this to mean that a plausible argument can be given for using, in a new situation, a generalization of a mathematical model that has previously been used successfully as a description for Physical phenomena. The form of a "plausible argument" is not given a priori, it's just a question of what Physicists as a group find plausible. Plausibility has an acid test, which is whether a given Physicist thinks an idea for a new class of mathematical models has enough promise that they spend their own time developing the mathematics and its relationships with experiment. All that said, QED is physically successful, and enough Physicists found it plausible to consider generalization to non-Abelian gauge fields that, over the course of 15 years, from the mid-1950s to 1972, say, with perhaps a few hundred people working on them, a new, moderately empirically successful class of mathematical models was constructed. The analogy with QED is significant, but it's the arguments for why someone in 1955 might think intensively about such models that I think you're not paying enough attention to. Those arguments are still known in the community, and they play out in various ways in the comments on your question and on your comments, but I think it's fair to say that they are not very clearly elaborated. There is no axiomatic QCD, that lays out both the mathematics and why it's especially natural as a Physical model, for a Physicist to point to, for example.
Underlying all your questions, answers, and comments, however, is your railing, as I see it, against renormalization. 40 years ago, you would have been in company with very eminent Physicists, but today the game has largely moved on. People do talk loosely about bare and real electrons as you do, but the principal modes of discussion are now in terms of the renormalization group and the surrounding mathematics, which is well enough constructed as mathematics that as far as I can tell most Physicists are content with it, and almost all Physicists are content to calculate with it. As far as I've seen, models for real, shielded particles are relatively ad-hoc, and in any case, and insofar as they are not ad-hoc, are ultimately so grounded in the Taylor series mathematics of Feynman diagrams and the renormalization group approach that they are essentially not a new approach.
All sorts of moves are being made, both in the system and on the periphery, to tighten up the mathematics more, or to construct new non-perturbative methods, but they will remain peripheral for most Physicists until an approach is constructed that is significantly better as mathematics than the renormalization group approach, which seems more explanatory, and which is, additionally, more usable in an engineering sense than the renormalization group approach. I think it's significant that the renormalization group approach is more-or-less usable as engineering, but I think it's clearly enough not easy enough to use the renormalization group approach as engineering that a replacement for this way of thinking and constructing models will emerge in due course. I conclude, therefore, that I agree with you when you write “I conclude that it is not the only possible way of description of interactions“, but the hard issue is how to construct something that is better, preferably significantly better. Gotta do the math, and it's gotta be good math, but, harder, because it requires a certain kind of simplicity that we'll only recognize when we see it, it's gotta be good for use as engineering as well.
Best Wishes, Vladimir!