The "Starfleet mostly sits in mothballs in peacetime" concept is IMHO dubious, because most of Starfleet's duties are peacetime ones, and Starfleet in peace is stated to be constantly short on ships (so that help always arrives on the eleventh hour if then, and comprises just a single starship).
We saw little evidence that older, TOS movie era ship types would serve in secondary roles in the TNG era. There was the Lantree
doing a supply run - but sister ships Brattain
appeared to be in frontline use. These latter vessels had higher registries, suggesting lower age; all the ships of this class seen fighting in the Dominion war had these higher registries as well.
Yet if there was a surge of ships for the Dominion war, activation of idled units is a more likely explanation than construction of all-new ones, as we virtually never saw ships with "wartime" registries, that is, registries higher than those of the prewar DS9 and VOY hero ships. And we never saw ships under construction in that era - only older ships under repair in the prewar flashback scene of VOY "Relativity". Arguably, starship construction is such demanding and time-consuming work that a mere half-decade war won't see significant numbers of new ships completed; peacetime pressures are already so high that there cannot be appreciable increase in production speed and volume in wartime; and a century doesn't make such a difference in starship technological capabilities that a major modernization of the fleet would be needed when entering a demanding conflict.
And a Federation with hundreds of member planets has only one?
It might be felt politically necessary to send the officers from all of these member planets through one and the same facility, to create proper ties of loyalty. And this one facility could be a very capable and capacious one: for all we know, the entire city of San Francisco is its campus, and attendance is in the high seven digits. We already have universities like that today, even if their facilities are slightly more distributed.