I do have an easier time believing there was a "communal mothball" owned by all of the Clans in the earlier decades of their existence. They inherited a crazy amount of equipment from the SLDF-in-exile. Enough to fill entire divisions and warship fleets. Brian Caches full of stuff, naval caches full of stuff. More than they could possibly use or need at that time. It's easier to imagine that there were caches of stuff that none of the Clans sought to own at that time.
The Clans were too small. Their toumans were a fraction of what we see in the 3050s-3060s, and their fleets were most likely also smaller. They were scaling up gradually those first few generations (with some noteworthy setbacks in cases where warriors adopted from the Pentagon populations revolted against the Clan way). Small number of worlds, small populations, small warrior castes.
The Mongoose absorption happened in 2868. The Clans were founded in 2807. So about 60 years into the Founding of the Clans. They were still smaller then.
As the Clans grew, larger populations, larger toumans, more worlds, they actually needed a lot of that cached stuff, or sought to own it themselves just in case they ever did. I suspect somewhere in there, they began to divvy it up more via votes or probably more commonly, Trials. Do what the Jaguars did but on a more routine basis, a bunch of jumpships/dropships here, a warship there, until there was little to nothing left as a "communal cache" and instead each Clan had the resources to maintain their own reserve caches.
I think that gradually more and more of the naval caches and brian caches of equipment were claimed by specific Clans, which were used to fill out the toumans, or sat in reserve but still owned by a specific Clan.
I've been going through Wars of Reaving again, looking at all of the naval cache references. Every one I can find has a Clan attached to it, ownership wise. Except for one reference used by the Cobras toward the end of this list.
-The Diamond Sharks got their Essex out of their Shark Babylon naval reserve cache and quietly snuck it back to the Inner Sphere.
-The Sharks activated an Samarkand-class carrier out of the Sharks Vinton naval cache. They noted the presence of two other vessels but lacked the resources to activate them.
-The Vipers pulled an Aegis out of their New Kent naval reserve cache.
-At Priori we have the Burrock rebellion, and they activated ships from the Adder's last naval cache there.
-The Coyotes found 3 Star League era warships docked together to make a crude space station in System 671-H. A bandit hideout basically. It concluded they were from an old Clan Mongoose cache.
Very notably, after the Wars of Reaving, there is talk of a shortage of naval equipment. The Adders, trying to rebuild, have nothing left in caches. There is no talk of getting anything from any communal caches. The Stone Lions work to activate 2 warships from a Hell's Horse cache. The Cobras activate two Sovetskii Soyuz vessels and tried to activate a Samarkand but gave up after years of trying and turned it into a training vessel. No cache name is identified for where these three vessels came from. It's the only unnamed cache reference with no Clan specific name attached to it. But its ownership doesn't seem to be in dispute, there is no talk of trials or fighting involved.
By Wars of Reaving Supplemental, it is noted that there is a shortage of transportation assets, jumpships and dropships, throughout the Homeworlds. But there is no talk of activating caches. Instead, the Coyotes have begun building new jumpships and dropships. It's implied that these vessels are appearing all over Kerensky space, suggesting that the Coyotes are exporting/trading them, or that the other Clans are trialing for some of them.