Shir Shel Yom

The Gra has a seperate shir shel yom for every holiday (based on maseches sofrim.)  According to his view that is the only song said.  Common practice is not like that, rather on Yom Tov we just say the regular song of the day.  The Magen Avrohom says in siman 132 the reason is since its just a remembrance to the mikdash we don't have to be so exact.  Many poskim advise not saying on Yom Tov שבו היו הלויים אומרים בבית המקדש for that would be a lie since they wouldn't be saying this on that day.  However, the Shmeras Shabbos C'Hilchasa still advises saying היום יום __ לשבת to fulfill the opinion of Ramban that mentioning the days of the week until Shabbos is a fulfillment of זכור את יום השבת. 

However, the Chashek Shlomo is bothered by the custom of the Gra to say only one shir shel yom, say two, that of the תמיד של שחר and that of mussaf?   The Magen Avrohom (ibid) is of the opinion that the assumption of the question is wring, on Yom Tov they only said the song of Yom Tov even by the תמיד של שחר.  This matter is disputed in other Achronim as well.  (See Presha siman 133 that actually suggests saying two.)   Rav Shlomo Hakohan explains the Gra's position also isn't to mimic the song of the Beis Hamikdash.  That was special for when our actions were desired by God, now we have no right to sing those songs.  What we say is a praise to Hashem for the daily creation, what is unique to that day and on Yom Tov the unique aspect of the day is the Yom Tov.  (What I like about this is that it explains how the Gra could have a special shir shel yom for Yom Tov shenei, in the mikdash it was chol, how could there be a special shir shel yom?  But if its just to mention the uniqueness of the day, its understood.)

Either way, everyone says on Rosh Chodesh ברכי נפשי, why is it unique?  I think the answer is because the Gemorah says we want to make known when Rosh Chodesh is, so we do many things to "announce" its presence.  However, what I don't understand is why Ashkenaz says מזמור שיר חנוכת הבית on Chanukah, which the Gra explains is the shir shel yom, why is it different than any holiday where Ashkenaz doesn't switch?  Maybe Ashkenaz isn't saying it as the shir shel yom, rather its part of the hodayah of Chanukah.  Meaning according to the Gra its the song of the day as opposed to Ashkenaz its meant as a song of gratitude.  (I don't know, its purely speculative.)

For those interested in explain of the mizmor and its connection to Dovid (and possibly Chanukah) see here.  (Thanks to my father for the mareh makom.)

As an aside, is it a song as the words shir shel yom indicate or was it just said as the words שבו היו אומרים indicates?  See here.

I saw someone ask according to the Gra that the shir of Rosh Chodesh trumps all, why is the haftorah this week that of Chanukah and not that of Rosh Chodesh?  I would assume that the haftorah follows the last laining which is that of Chanukah but see Magen Avrohom 684:2.

P.S. Sorry for the lack of mareh mekomos, I'm pressed for time and can't find the location of everything immediately.