There are billions of dollars (face) of junk silver that can be melted into bars, rounds, and American Eagles. Expensive to melt and process, so they won't do it unless they have to. The big dealers like APMEX can pay mints to mint new rounds and bars a lot cheaper than they would have to pay the public. The mints can get all the silver they need without going to the public.
The mints are working at whatever capacity they have.
It is not expensive to get rounds minted. The old man Pawn Guy did it... last time I checked it was .59 per round but that depends on purity of the bars you give them.
The us mint is different, they do not make blanks, that is contracted out based on demand estimates and have a long lead time to get extra ones.
Junk silver is not .999 and while there is some above spot it is not melted down anymore
Dealers near me have always paid over spot for eagles
Only a hundred million new worldwide investors buying 10 ounces a year would alleviate your fears. That would exceed total mine production as it stands now and silver isn’t a corn crop. A lot of it ends up in landfills every year. So lets do everyone a favor and help drive up the price so people could recycle those discarded products in order to end that practice. There is only so much silver to be mined.
Btw, if the miners increasingly sold their product directly there would be reduced Comex inventory for end user manufactures. Then they would be forced to sidestep Comex and enter the free market. That would be quite a site!