My take on the paradox of thrift:
It is correct that when people choose to save $100 it does deprive someone of $100 income, with savings remaining unchanged. The problem isn't that this is incorrect, but that it's irrelevant.
When someone forgoes spending $100 they're signalling a preference to the market, that the market is over-producing. By saving $100 the person tells the market that the total value of production is $100 less than it was before ($100 less value to the consumer).
The result of this is that the market reallocates resources to bring that $100 value back to where the consumer wants to spend again. The Keynesian view is that this means something is broken, that "aggregate demand" will continue to drop or stay down forever. It won't, it's how a healthy economy works. The Keynesian response is to print or borrow money to get the consumers spending the $100 instead of saving. The problem is that all the Keynesian has done is reduce the value of money by $100, not raise the value of production $100. The same problem persists.
There are a number of things that cause consumers to value production less, all of them exogenous. Things such as poor fiscal/monetary policy and technological shifts. All Keynesians are doing is glossing over the problem. The adjustment will happen eventually, they can't stop it.