I figure the answer is hidden in the definition of "marginal contribution to risk". I try to use the notation of the paper you linked.

Marginal contribution to risk is defined as:

$M_i = 2C_{ip}$ with $C_{ip}$ being the covariance between asset $i$ and the portfolio $p$.
One can now argue that, since this covariance also depends on the other assets $j\neq i$, the value $M_i$ is implicitly connected to the other assets via the correlation structure.

On the other hand if all assets are *uncorrelated*, we have that $C_{ip}=C_{ii}w_i$, since $C_{ij}=0$ for $j\neq i$. Thus, the marginal contributions do *not* depend on the other assets. Therefore one can speak of a more "pure" contribution to variance in this case.

Finally, if the asset returns are *independent*, they are also *uncorrelated*.