The big companies don't care if some great candidates can't do these kinds of problems, there are many others who can. They care about not hiring bad candidates

The big companies don't care if some great candidates can't do these kinds of problems, there are many others who can. They care about not hiring bad candidates, period. Anyway, I've done both types and for me the false positive rate for take-home assignments is far greater. I've seen people ace home assignments and later turn out to be mediocre engineers. Maybe they were cheating. Maybe not. But I've never seen anyone really ace an on-site coding question and not being great (I'm sure there are some, this is just my own experience).