= Solution
For a mixed spin-half state, the <Bloch vector> formula gives
$$
\boxed{
\rho=\frac12\left(
I+\frac2\hbar\langle S_x\rangle\sigma_x
+\frac2\hbar\langle S_y\rangle\sigma_y
+\frac2\hbar\langle S_z\rangle\sigma_z
\right)}.
$$
Physical consistency is equivalent to the resulting Bloch <vector> having length at most one.
Three linearly independent Hermitian operators do not always suffice. The operators
$$
I,\quad\sigma_x,\quad\sigma_z
$$
are linearly independent, but the first expectation is always one and none detects the $y$ component. States with Bloch <vectors> $(0,c,0)$ and $(0,-c,0)$ have identical expectations for all three. The correct criterion for <informationally complete three-observable qubit tomography> is that the three traceless parts span the Pauli space.
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