The silver ETF SLV is basically like a stock. There are a certain amount of shares of SLV on issue, each one is backed by a certain amount of physical silver, sitting in a vault.
When people sell the ETF SLV they are selling the shares to someone else, the ETF doesn't then go sell a whole bunch of silver. When the price of SLV moves too far from it's NAV arbitragers are ready to keep the price in line.
example; Massive demand for SLV shares for some reason, drives the price up 10% above NAV. Arbitragers short the ETF, buy physical silver, deliver it to the ETF trust which then issues NEW SLV shares, the arbitrager closes out the position at profit of the premium till SLV trades back at NAV.
When people were dumping the ETF's as you say, they were simply selling their 'shares' backed by a certain amount of silver. The ETF trust did nothing. Arbitrages saw that SLV shares were trading at a discount to NAV, they buy SLV shares, short physical silver, take the shares to the trust, the trust destroys the shares and hands over the silver and they close out their physical position.
The arbitrager's had to deliver physical silver to the vaults of the SLV ETF to receive the new SLV shares, it could not have been 'paper silver'.
The fact that SLV trades so close to it's NAV is testiment to the fact that there is plenty of physical silver available in the market at spot when needed.