Since doing this research, I came across this article from my old yeshiva, which essentially makes the same points.
Here is the link - the quoted text (everything below) is at the end of the page there
On Rosh Chodesh we take off our tefillin before musaf for one of two possible reasons:
(1) The Beit Yosef say that it is because in musaf we say "keter" in kedusha and it is not appropriate simultaneously to wear the "crown" of tefillin; or
(2) The Radbaz (responsa vol. IV, 80) and the Levush (423) explain that since musaf is in place of the musaf offering of the day, and since Rosh Chodesh is similar to Yom Tov (when tefillin are not worn), we refrain from wearing tefillin at least during the time which corresponds to the time of the sacrifice.
One possible difference between these two would come to light in the case of one who does not say "keter" in kedusha. However, see the Rema who writes that such a person too should nevertheless take off his tefillin before musaf (see also M.B. 25:62).
If one started to pray musaf, forgetting to take off his tefillin, see M.B. 25:61 (although the Zohar in Pinchas warned that one who prays musaf with his tefillin on is subject to a very great punishment).
When should one take them off on Rosh Chodesh?
The Mishna Berura here (25:59) and in Hilkhot Rosh Chodesh (423:10) brings several minhagim: (1) After the kaddish which precedes musaf. (2) Before musaf, in "u-va le-tzion go'el," right before "Yehi ratzon ... she-nishmor chukekha" (in order not to interrupt between kaddish and musaf). (3) One should take the straps off his finger before "Yehi razon" and take the tefillin off after kaddish.
It appears to me that most people follow option (1), but each individual should gauge his own dexterity in the removal of tefillin and make sure to have them off in time to begin musaf with the congregation.