I think the real answer to your question is politics. Helicopter money is a political question in a way that QE is not. The question of what social programs we have, and what they cost, who gets them, etc., has been a major issue already in the 2020 election, and people are already drawing parallels between cutting checks in this crisis and a UBI for example.
The Fed can do this at all right now because we created a body of subject-matter experts and deputized them to pull the trigger without having to check with anyone. This is a rare thing in a democratic government, and is possible mostly because we have given them boring powers that only wonky people bother to even read what they are. Even then, a few people read what they are and complain about them, so even that is a bit perilous.
If you wanted the Fed or a Fed-like mobilization for the helicopter money you first need to make the process of getting $1000 boring, or wonky, or at least politically uncontroversial so that people accept an unelected body of bureaucrats outside the government process doing it unilaterally in the middle of the night without telling the rest of the government. I would not rule out those conditions being the case at the end of this crisis, but they aren't the case at the beginning.
But the way we solve political problems (or, more recently, the way we don't solve them) is through Congress, which is an inherently slower approach.