Okay, so here's my understanding of it, and this is drawing on experience working for a silicon polymer chemist, so hopefully I don't lead you astray.
The oxygen could for a moment in time obtain the electrons, and it's quite possible it does, but it will not as readily grab the proton as nitrogen will. Often it will just quickly kick the alcohol back out and return to it's original state. Now, that being said, when the nitrogen eventually does hold that negative charge, it will be much more likely to make a nucleophilic attack and grab the proton as seen in the polymerization reaction you have outlined. Mechanisms are just the way organic chemists believe electrons moved based on the final product observed and what steps would result in lower energy states or move the reaction forward via Le Chataliers principle.
What I can tell you for sure is that how these two molecules approach each other (through the medium being used) and their orientation in space, as well as so many other reaction conditions (like temperature, length of reaction time, etc., which I'm not aware of for this problem), could contribute significantly to how the polymer is formed.
I really hope this helps you guys out! And please remember, the true professionals on this forum are Dr. Jim Romano and Nancy Steen. This is just the way I view the problem presented and hopefully it makes sense 🙂