in the end it seems to me that the definition of ergativity in terms of intransitive/transitive subject and object is hardly satisfying and looks either circular or ethnocentrist. So why is it still in use? And more generally why isn't the fuzzy and dubious notion of subject made obsolete in linguistics and replaced by more rigorous and appropriate concepts when dealing with languages in general?
This paragraph seems to suppose that in all languages that display the phenomenon of "ergativity", the concepts of "subject" and "object" are artifacts of applying an analysis based on non-ergative languages. This really isn't the case, though: many languages have ergative morphology and also other aspects of grammar that make it possible to demonstrate in a rigorous fashion that absolutive-marked NPs in intransitive sentences and ergative-marked NPs in transitive sentences behave alike in cetain contexts or in certain respects. That is, there are motivations for distinguishing the concept of "absolutive-marked noun phrase" from the concept of something we can call the "subject" that arise from the structure of these languages, not just from the imposition of external concepts of subject- and object-hood taken from languages without ergative morphology. Sure, we could call this concept something other than "subject" if we wanted to, but "subject" seems to fit the semantics and has mnemonic value, so why use a different word?
In many (but not all) languages with an "ergative" system of case-marking, the noun phrase that is morphologically marked as ergative case in a transitive clause acts syntactically similar in a number of ways to the noun phrase marked as absolutive case in an intransitive clause. This phenomenon is often called "split ergativity".
So within many particular languages that makes use of ergative marking, we can have quite clear evidence of a concept of "subject" in those languages from other patterns of marking in other contexts (e.g. the difference in case marking in the past/perfect and present tenses in Hindi and Urdu), or from syntactic evidence e.g. the behavior of "pivots".
As for the more general typological criteria for applying the concept of "subject" and "object" to a language that only uses ergative morphology and has a fully ergative syntactic system, there might be some fuzziness. I haven't studied the matter, so there might be more subtle grammatical phenomena other than morphology and pivots that we can use to identify the subject. If not, it just means that "subject" and "object" are defined at this level by mainly semantic grouping, which is not that strange for universally applied typological categories. We can distinguish them by the same vague kind of criteria that we use to distinguish "nouns", "verbs", and "adjectives".
At that level of typology, these kind of "universal" classifications are often disputed if we take them to represent some important fact about the structure of all languages, but they are often felt to at least provide some rough method of inter-linguistic comparison. It's easier to remember a few terms that mean slightly different things in different languages than it is to remember a bunch of terms that are all specially designed to refer to specific features of individual languages.