It had more to do with the fact that Berman doesn't like having ships of the same class as a "hero" ship from another series showing up randomly on another show. If it was clearly stated that the model was of a different starship, like the Oddessey or the Bellerephon on DS9, that's one thing. But if it's seen in the background for a few seconds, say a Sov during "Sacrifice of Angels," you would have had 87 internet posts titled "ENTERPRIZE-E IN DOM WAR!" showing up the next day.
That said, B&B eased off on that rule in the later days of DS9 and VOY, hence so many Galaxies during the war and the final episode of VOY. Interestingly enough, if the Federation held the Sovs back as a last line of defense post-First Contact, wouldn't one of them responded to the appearance of the Borg ship in "Endgame"? As I noted in my fiction and in the background text I wrote for GaW (which I hope one day will use this and many of Redragon's models), the Sovs are expensive to build and during the war, it made more sense cost-wise to build more of other classes of ships than one or two Sovs. That's why there wasn't more than say four or five by my reckoning during the time of SFC3.
Edit: Also remember that there weren't more than six to twelve Galaxy-class ships, as well. If the Sov is meant to replace them, one would think that there would by that many or less.