That's the point of the fiscal multiplier figure: according to their estimates it improves the economy by a multiple of 2.6x the cost of the money transferred.

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Seems like reverse "trickle down" in another respect too: the poorest recipients got better off but the gap between rich and poor still widened, presumably because the poor spent much of their increased budget buying things from the rich.

It wouldn't surprise me if that were the case.

Imagine someone who sewed things by hand, given (or even loaned) the money to purchase a sewing machine, making them many times more productive. Now the community can have things sewn faster and more easily with less effort, with tons of downstream effects (more sails, better clothing, stronger soft goods, etc) and it's easy to see how the net improvement to that community can far exceed the price of the machine.

Another way to ask that question is "does there exist tools whose price is less than the total amount of increased economic activity they enable", which when phrased that way, it seems clear that the answer is "yes".

I would think that the more consumers that can "vote with their wallet" on which products are the best, the more quickly the market will identify and reward good products. Identifying and rewarding the best products (i.e. those processes that create the most valuable outputs from the inputs) is one of the primary benefits of market-based economies. The positive feedback loop of reinvesting profits helps them lift above the general noise.

Yes now go ahead and pay out the benefits in a demurrage currency so that benefits payments can't be withheld from the economy which means the amount you need to inject will be much lower.

The people who get their tax reductions tend to be rich and not spend their money.