The explanation offered:
"In our Gemara, the residents of the courtyards that open onto a
Mavoy bought a barrel of wine together and each resident's portion is not
determined at the onset of Shabbos. When they apportion the wine at a
later point, they will be relying on Bereirah to let the Shituf work by
determining retroactively who owned each portion of the wine at the onset
of Shabbos."
- leaves me wondering: after all, if each resident brought his wine flask and poured it into the Eruv cask before bein-hashemashoth the situation is also that "each resident's portion is not determined at the onset of Shabbos" because the portions are all mixed together!
a) It seems to me that the basic point is that to make an Eruv each member must contribute some which is definitely his. Since the partners in the wine-barrel never divided it up, we say "Ein Bereira", and no one has given anything recognizable as belonging to him to the Eruv-barrel. Hence we say the Eruv is invalid.
b) It would also seem reasonable to explain why Haza"l used Ein Bereira here in this unusual way (see Tosafoth) is because the whole point of making an Eruv is psychological, to make an impression on the minds of the dwellers in the courtyard; using a common barrel of wine which was never divided means in effect that only one person - the one who purchased the common barrel - need be actively involved in the making of the Eruv, thus the impact of actually making a common kitty of food for Shabbath would be missing.
Yeshayahu HaKohen Hollander
a) There is a difference between when ownership has not been determined at all, and when it was once determined but presently we do not know who owns what (because it all got mixed together). The Gemara is referring only to when ownership was not determined at all. If it was determined but is no longer recognizable, the Eruv would be valid.
b) Your logic is good. However, if all of the members need to be involved in the Eruv as you say, how can one person be Mezakeh for everyone else without informing them (see Daf 79b and 80b)? The other residents were not actively involved! It must be that whenever the ownership of each portion is not known, there is no clear indication of "Eruv Da'atan," that they consent to each other and are not Makpid on one another (49a), since there is no portion that is independently owned by each of the residents of the Chatzer. However, we assume that each resident will not be Makpid in a case of Zikuy. (Perhaps this is your intention as well.)
Kol Tuv!