K (ngn/k), 16 bytes
{(&*/'2\'!#x)_x}
A near-backport from my own J solution. Instead of checking each 1-based index has exactly one 1 bit, this checks if each 0-based index is all ones in binary. This works in K (2\0 is an empty array and its product is 1) but not in J (#:0 is 0 instead of an empty array).
K (ngn/k), 17 bytes
{(|1_(-2!)\#x)_x}
Similar to Jelly (x_y cuts y at the indices specified in x), but had to be careful to avoid generating large indices or indices in non-increasing order (which give domain error).
My initial approach used 2\ (integer into binary vector) and 2/ (binary vector into integer), but I realized that simply repeatedly dividing the length by 2 gives the correct cut indices.
{(|1_(-2!)\#x)_x} Input: a string of length 2^n-1 (example: "abcdefg")
( #x) length of x = 7
(-2!)\ repeatedly floor-divide by 2 until convergence and collect:
(7 3 1 0)
1_ remove head which is too large (3 1 0)
| reverse (0 1 3)
_x cut x with those indices ("a"; "bc"; "defg")