Cases where TDA outperforms public benchmarks? Precise Question
What are some specific examples where topological data analysis (TDA) outperforms other models on publicly available data?
Context
When new ML algorithms are developed, it seems common practice to apply them to publicly-available data and provide accuracy comparisons in order to illustrate the novelty of the algorithm in a scientific way.
As a concrete example, consider the KDD cup references in this XGBoost paper http://dmlc.cs.washington.edu/data/pdf/XGBoostArxiv.pdf
 A: Comparisons between ML-at-large to topological data analysis (TDA) is difficult. This in part because much of what TDA attempts to do isn't even on the menu for most ML projects. One place (other than article databases of course) to look for examples of TDA is the Applied Algebraic Topology Research Network's website, or their YouTube channel.
I am relatively new to TDA, and primarily focusing on persistent homology at this time, but I will offer a purported example. In a talk given by Renata Turkeš titled On the effectiveness of persistent homology they describe a comparison of deep neural networks vs other techniques including classic persistent homology algorithm in predicting the correct homology class, and the latter was non-dominated by the deep neural network architecture.
Such comparisons only look at a small fraction of the possible models, datasets, and other choices. So arguments over the superiority of broad frameworks are often quite drawn out, and quite uninteresting in my opinion. There is almost always a Pareto front lurking nearby such debates. But if you want to get into the tradeoffs of the comparisons that were made in this particular case, I would recommend reading Turkeš 2022 to dig a little deeper. Figure 3 is the bar plot they showed in the talk.
A: As of 2016, it seems difficult to concretely answer the question. Perhaps an ambitious, risk-seeking, grad student will provide more insight on TDA's performance as compared with well-known benchmarks. 
