I don't think it's particularly controversial to say that the "standard" way people use to compare quantum states is via the fidelity. Yes, sometimes the trace distance is used as well, but it seems to me that the "canonical choice", the first one in most people's minds, is to use the fidelity.
There are several ways to give an operative meaning to the fidelity, but one that seems particularly fitting in this context is the observation that the fidelity $\mathrm F(\rho,\sigma)\equiv\|\sqrt\rho\sqrt\sigma\|_1$ between equals the Bhattacharyya coefficient between the probability distribution resulting from measuring the states, maximised over all possible POVMs producing said distributions. Some discussion about how to interpret the classical Bhattacharyya coefficient can be found in this stats.SE post.
On the other hand, the trace distance $D(\rho,\sigma)\equiv\|\rho-\sigma\|_1$ can also be given a rather direct operative interpretation, in that it quantifies the optimal probability to discriminate between the two states.
If the goal is to compare how close a state is to another, I would have intuitively guessed the latter choice to be more "natural": I'm quantifying the maximal probability to tell apart the two states. The Bhattacharyya coefficient, essentially the inner product of the square roots of the probability distributions, also has some geometric meaning, and I'm vaguely aware that it's related to the Fisher information or other metrological quantities, but I don't know how to directly directly connect this with the goal of quantifying closeness of states.
So, is there a clear way to justify the usage of the state fidelity over the trace distance from an operative point of view? Of course, this might also depend on the context and for what purpose the states are being compared, and then I'm asking if there is a clear way to distinguish between situations where the fidelity is the "right" choice, and situations where the trace distance is.
