rosh chodesh Teives / Chanukah -- 1 chiyuv to say full hallel, or 2 independent chiyuvim, full hallel + chatzi hallel

When we daven b'tzibur the custom is for the chazan to say out loud the last sentence or two of each section of hallel, similar to other parts of davening when the chazan reads aloud the last line of a bracha or section of tefilah. When chatzi hallel is recited on rosh chodesh, that means the chazan says the concluding sentence of the paragraph that starts "Hashem zicharanu yevareich..." and the concluding sentence of the paragraph that starts "mah ashiv." When we say full hallel, the chazan also says the concluding lines in the paragraphs before these as well, i.e. the paragraph that starts "lo lanu..." and the paragraph that starts "ahavti..." 

I don't understand why we do things this way. "Lo lanu" is the start of a perek.  That perek concludes "v'anachnu nevareich K-h." Why doesn't the chazan read straight through the entire perek and only say out loud that last line, not the last line before "Hashem zicharanu yevareich...?"  "Hashem zicharanu yevareich..." is an independent unit only when we say half hallel because when we say half hallel we skip "lo lanu" and start in the middle of the perek.  But why create that artificial break when we are saying full hallel?   

The simple answer of course is that the chazan is just following the pattern he is used to from the times we say half hallel. But maybe there is more to it than that. 

I saw the following chakirah: when we say full hallel, is there also a chiyuv of half hallel lurking in the shadows, or does the chiyuv of full hallel eclipse the chiyuv of half hallel and so that doesn't exist at all? For example, on rosh chodesh Teives, are there two independent chiyuvim of hallel, i.e. a chiyuv of full hallel because it's Chanukah, and a chiyuv half hallel because of rosh chodesh, or would you say that since there is a chiyuv full hallel, there can't possibly exist a separate chiyuv of half hallel? 

Nafka minos: if someone only knows part of hallel, should they at least say that, even if they don't know the whole thing? If someone said a bracha and omitted a part of hallel, is it a bracha l'vatala?

The same chakira can even come into play on a day like the first days of Pesach. Is there only a chiyuv of full hallel -- all or nothing -- or can you argue that the first days cannot be worse than chol ha'moed, where there is a chiyuv of half hallel?  Just because there is an additional chiyuv of saying full hallel does not negate the lesser chiyuv -- or does it? 

Minhag Sefard is to say the bracha "ligmor es ha'hallel" when reciting full hallel and "likroh" when reciting half hallel, but minhag Ashkenz follows the Maharam Rutenburg who writes that we should always say "likroh." Tur (488) explain that Mahram Rutenburg was concerned lest one leave out a word in hallel, which would render saying "ligmor" a bracha l'vatala. Mishna Berura asks: if the mitzvah is to say full hallel, even if one said the bracha "likroh," wouldn't it be a bracha l'vatala anyway?  Since you missed a word, you didn't do the mitzvah, and the bracha is therefore l'vatalah?  

The MB's question hinges on the chakirah we raised. If there still exists in the shadows a chiyuv of half hallel even on days when there is a greater chiyuv of full hallel, then if one missed a word, one would still be yotzei the lesser chiyuv of half hallel and the bracha would be chal on that chiyuv.  MB must have assumed like the other side of the chakirah, namely, that when there is a chiyuv full hallel, it eclipses and negates the chiyuv half hallel. 

I think the nusach ha'tefilah indicates that even when there is a chiyuv to say full hallel, the chiyuv of half hallel still exists in the shadows and that's why the chazan still sticks in those breaks in the middle to demarcate the same section breaks that are used when we say half hallel.