I don't know why this oversimplified rethoric is so popular. No, nobody is dumping free money into the stock market. What happens is that the money is taking a very indirect route through the economy and then it slowly ends up with rich people.
I swear Austrian economists have rotten people's brains with their cantillon effect rethoric. They never described the mechanism that somehow wires the money straight to the bank accounts of the rich, they simply assume that this happens without any evidence.
What really happens is this: Lindner introduces a so called vehicle fuel rebate. This fuel rebate lowers the cost of fuel by 35 cents. Gas stations pass some of the rebate but not all of it onto the customer. If the price was kept identical, then demand wouldn't change but if the company lowers the price, there will be additional demand and it will end up with more profits overall. In other words, some companies have a degree of pricing power that they use to increase their profit margins.
When the money supply grows, even if it was paid out fairly to everyone as a UBI, it would slowly end up in the hands of companies with pricing power and this pricing power has absolutely nothing to do with the money that is being used. A gold standard or Bitcoin wouldn't save you from this. It will just mean they have all the Bitcoins or all the gold.
Edit: I would also mention that in a zero lower bound system, it is impossible to pay off all debts without debt deflation which in turn means that you would have to design your monetary system around the expectation that not all debts will get paid and that you will have to do regular debt jubilees. The real world isn't a permanent competitive equilibrium.