building a shul -- mitzvah or hechsher mitzvah?

Our parsha discusses the mitzvah of binyan mikdash.  Is there any mitzvah to build a shul?

The Rambam writes in Hil Tefilah 11:1

כל מקום שיש בו עשרה מישראל, צריך להכין בו בית שייכנסו בו לתפילה בכל עת תפילה; ומקום זה, הוא הנקרא בית הכנסת.

From the Rambam is sounds like building a beis knesses is only a hechsher mitzvah.  If you have 10 people, then you need to set aside a place to daven; however, if not for tefilah b'tzibur, then it sounds like there would be no mitzvah.

Once upon a time we discussed the first Rambam in Hil Beis haBechira:

 מצות עשה לעשות בית לה', מוכן להיות מקריבים בו הקרבנות, וחוגגין אליו שלוש פעמים בשנה--שנאמר "ועשו לי, מקדש"

Here too, the Rambam's language implies that it is the need to offer korbanos that is the driving force behind building a mikdash.  That being said, there is still a mitzvah to build a mikdash independent of the mitzvah to offer korbanos.  In no way is one chiyuv dependent on the other. 

Perhaps with respect to building a shul as well, although the Rambam sees tefilah as the driving force, the building still counts as an independent chiyuv.

One possible nafka minah: According to the Rama that allows amira l'aku"m m'makom mitzvah, can you allow non-Jewish workers to build a shul on Shabbos?  If building a shul is only a hechser mitzvah, perhaps not (Machtzis haShekel on MG"A 276:8).

See Rav Wahrman's She'eiris Yosef vol 5 for more.