What the Fed accomplishes by handing out the $1 trillion is this: they kept all the _other_ money circulating. If people wanted to hoard $1 trillion because they're scared about the future, well, now they've hoarded it. They feel safe and they spend the rest of their money, and the economy keeps humming along.
If instead they'd tried to hoard that $1 trillion without Fed intervention, there'd be an economic collapse.
Sure, you can give all the money directly to the common people, but you'd end up having to do that forever since the wider economy would collapse as rich share holders first feel doubt, then have those doubts confirmed in a self fulfilling prophecy.
Common people should have non emergency sources of welfare, and governmental aid to rely on. If they don't, then the situation is aleady dire.
The pandemic is showing that most people and state governments are willfully unprepared to provide such welfare even in an emergency.
But you're right insofar as the Fed is trying to change inflation expectations by printing money. But so far they've failed because no one believes they'll actually let inflation happen once the economy recovers. Hence Jerome Powell's recent statements about the Fed changing how it trades off inflation against economic growth. It's a way to more credibly commit to letting inflation actually happen.
Didn’t the fed announce they would run inflation a bit hot for a while just last week?
So someone, somewhere, is sitting on cash, which is deflationary and not what you want in a weak economy.
The cure for that problem is cheap money plus expected inflation (cheap money so people can get cash, expected inflation so they go spend it). The Fed can make money cheap, but it can't force people to expect inflation. It can do stuff to encourage that expectation, though, and yeah it's doing just that.