There is no basis for it in the Orthodox phronema (mindset), because our understanding of the original sin is not Augustinian; never was, which is at the foundation of that dogma. We do agree with our Latin brothers that she was immaculate but was not conceived that way.
...why do these differences appear to be "no big deal"? I know you're not miles apart, but I just would have guessed that beliefs on Mary would be pretty important
It is important and it used to be a subject of heated debates among us and Catholics. However, we have come to understand that such theological issues can be resolved only in a a general or ecumenical council that is yet to take place after more than 1,200 years.
We recognize and know our differences in teaching and we also recognize that it is no different than, say, some fathers teaching filioque or original sin (i.e. +Augustine) and parts of the Church not being in agreement.
Until such a Council meets, we have agreed and the Pope has been instrumental in this, along with a general movement towards Patristic mindset in the Western Church, to concentrate on our common theology (which is 99%) and leave the difficult issues that divide us still to a future Council which, by the way, cannot take place until the role and scope of papacy is agreed upon (under current discussion between the two particular Churches).
There has been a slight set-back in this when recently the Pope decided to drop his title as the "Patriarch of the West" whereby such a move indicates that he is "outside" or "over" other Patriarchs, not as one of the Patriarchs, among whom he is senior in dignity, but rather as the Patriarch of the Church in title and not in honor.
I have not eard much more on this since then. However, if that is an attempt to place himself above other Patriarchs, the talks are doomed.
Thank you for your whole answer. Please forgive my ignorance, but is there one man who is the head of your Church? If such a future Council were to be called, who would be contacted on your side to set it up, and who would participate on both sides?