You should take the time to learn about the velocity of money. This wikipedia article does a good job of summing it up: https://en.wikipedia.org/wiki/Velocity_of_money
But basically, money doesn't cease to exist once you spend it. If Uncle Sam gives me $1000, and I then buy something from you, we have just generated $1000 worth of economic activity.
But then you spend that same $1000. We have now generated $2000 worth of activity.
Which is spent and respent very quickly, stimulating the economy.
So, even the "frivolous" spending benifits the economy.
The thing about the poor, when they get money they spend it.
Rich people, however, don't spend money. They're kinda famous for it. In fact, we know that if someone had more than $3K in their checking account, they didn't generate any more economic activity than they would have normally. They literally didn't spend it.
This research paper estimates that half of the 1st stimulus was spent within 10 days, and the remaining amount spent in the 2nd 10 days. https://www.ucdavis.edu/curiosity-gap/did-1200-stimulus-payment-boost-economy/
So, yes, giving the money to some people is a waste. Just not the people you thought.
And even if it didn't generate economic activity, giving the poor and jobless money so they can buy food and keep their shelter in the middle of a pandemic is the only right thing to do.
They're people. And they deserve life, liberty, and a warm place to sleep at night.
And, honestly, making the argument on a liberal board that $1400 isn't enough, you're only going to find agrement. Because we think $1400 a week until this is done would be just right.