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Quantum spin Hall (QSH) insulators have unique electronic properties, comprising a band gap in their two-dimensional interior and one-dimensional spin-polarized edge states in which current flows ballistically. In scanning tunneling microscopy (STM), the edge states manifest themselves as a localized density of states. However, there is a significant research gap between the observation of edge states in nanoscale spectroscopy, and the detection of ballistic transport in edge channels which typically relies on transport experiments with microscale lithographic contacts. Here, we study few-layer films of the three-dimensional topological insulator (Bi$_{x}$Sb$_{1-x})_2$Te$_3$, for which a topological transition to a two-dimensional topological QSH insulator phase has been proposed. Indeed, an edge state in the local density of states is observed within the band gap. Yet, in nanoscale transport experiments with a four-tip STM, 2 and 3 quintuple layer films do not exhibit a ballistic conductance in the edge channels. This demonstrates that the detection of edge states in spectroscopy can be misleading with regard to the identification of a QSH phase. In contrast, nanoscale multi-tip transport experiments are a robust method for effectively pinpointing ballistic edge channels, as opposed to trivial edge states, in quantum materials.