In Skin Game Lasciel told Harry that the coins weren't meant to be "locked up" very long. I forget her exact language. It was something about how corruption can't be kept hidden. I suspect the same applies to Demonreach as the Church keeping the coins in hiding.
Also, it is possible that whatever rules Demonreach operates under might make it impossible for the island to hold the coins. Maybe it could hold Nicodemus or another denarian but the coin might not be able to enter the vault with the coin holder. I'm just speculating, but I wouldn't be surprised if this or some other mechanism prevents the island from imprisoning the coins.
We've never seen Alfred act on its own volition to imprison someone. At the times when the Denarians were on the island, there was no Warden to give the order to imprison them. Almost all the creatures and beings in that prison are there because someone (i.e. the White Council I'm pretty sure)
put them there.
Could the prison hold a Denarian? I imagine so. Could it hold a Coin? I'm not so sure. If nothing else, there would be the risk of infernal intervention.
I think there are a couple of problems with this idea. First, the island doesn't like visitors and there isn't a practical way to keep an eye on the island other than being there.
I have to agree with Mira on this. Maybe Alfred doesn't like visitors on the island or maybe not, we just don't know. Apparently someone was able to operate a cannery there for a little while at least, so it's certainly possible to visit it. Alfred might not like Rashid, but Rashid still visited the island.
So I would imagine that the Council could send a party of Wizards to check the place out every so often.
And even if that isn't a viable option, it would not be that hard to keep track of who visited. The Council has vast resources of money. They could put a few boats on Lake Michigan in the area, disguised as ordinary fishermen or pleasure boaters or whatever to keep track of the waters around the island, they could hire someone to run a radar set on such boats to watch the skies about. It just wouldn't be that hard for the Council to keep track of comings and goings, at least in the real world. We don't know how hard it would be to access Demonreach through the Nevernever, but Nicodemus seemed to think it would be...difficult. And I suspect the White Council has the resources to keep track of the Nevernever near Demonreach, too.
It really is strange that the Council doesn't seem to be doing that, considering what that place is. Maybe, maybe, they're afraid of calling attention to it by watching it too closely...but I don't really buy that.
Mira has a point, this is really strange.
I suspect the most important problem is, the White Council doesn't choose who becomes the island's Warden. I'm pretty sure it's Alfred choice. According to Harry, performing a Sanctum invocation is dangerous, so it isn't something most wizards ever do. So the Council would be hard pressed to find volunteers to attempt to do what Harry did.
I suspect the Council usually does choose the Warden, when there is one. They might have to be careful to choose somone acceptable to Alfred, but I doubt that's an insuperable barrier.