The Treasury has the power to mint coins, if what I read is true. Now, I don't know that they should really want to do this, but it may all be legal and above board. If all of this is indeed true, then all we have here is a lot of huffing and puffing over nothing.
Now, it looks as if the government has set up accounts for Social Security Trust Fund. Are checks drawn on this account? If so, how can checks not be sent out if there's a government shutdown?
Nevertheless, the government could move the account to the Federal Reserve and not change anything at all.
The monthly checks that pay benefits would go out each month and it would be drawn on the Federal Reserve, not the Treasury.
The Federal Reserve is not the government. It is a government sponsored entity, so that is a distinction. It may be easier to make that distinction if the checks were coming from the Federal Reserve instead of the Treasury.
Update: Here's the chart with some comments attached.