Dear Jerzy,
You put your finger on some very subtle issues the importance of which I am afraid will escape many readers because it is rare for a person who by training and vocation is so steeped in a particular methodological paradigm to be able to take a step back and compare it to others. I'd like to laud you for that.
I happened to recently take a philosophy course called "The Scientific Revolution" dealing with the history and philosophy of science between 1543-1726. Prior to that, I was aware of some of the methodology of Aristotelian physics, but it seemed altogether very foreign to me. However, through that course I gradually came to appreciate that from within the Aristotelian worldview, his methodology was exactly the one that would have seemed relevant. There is a less well-known paper by Thomas Kuhn, entitled "What are scientific revolutions?" (written almost two decades after "Structure") which, if you are not already familiar with, you might find interesting:
http://www.units.miamioh.edu/technologyandhumanities/kuhn.htm
To very briefly summarize my own view on the issues you raise, I think you are correct that our present methodology is entirely on the descriptive side which misses any questions about the essence of things. While I would definitely not advocate a return to Aristotelian methods, I do believe that it is possible to extend the current methodology to arrive at something that is in a certain sense intermediate, by extending the expressive power of mathematics. I believe such an extension would be most usefully applied to that aspect of the essence of things we call ontology, thereby wresting it as a subject of inquiry from philosophy to physics.
In the abstract this is probably not giving you a good idea of what I am talking about, but my current entry to this contest contains a concrete approach to implement it, and I would very much welcome your constructive criticism on it, as it is a work in progress.
Best wishes,
Armin