Is it true? Ask Jelly!
Jelly, 3 bytes

ṭFẠ


F flattens the input list.

ṭ tacks on the original input list as an element, which is falsy if and only if it is empty.

Ạ then checks if any element in the flattened list, or the original list itself, is falsy.



(Original answer)

FẠ^Ṇ


Thanks to Dennis for encouraging finding a solution matching his.

FẠ gives 0 if the input contains a falsy value at any depth, else 1. This is what Ȧ does, except for empty lists.

Ṇ gives 1 if the input is a falsy value, else 0. The only falsy list is the empty list.

XOR-ing the two gives the answer.



F;WẠ


This is much in the same spirit as Dennis's F;LẠ, but instead of using L to put a zero in the list when the list is empty, it uses W to put the empty list into itself (producing [[]]), making it contain a falsy element.