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The two-dimensional electron gas (2DEG) that forms on a semiconductor surface can be used to explore a variety of phenomena in quantum physics and plays an important role in nanoscale electronics, such as transistors. Controlling its formation is, thus, relevant. Using angle-resolved photoemission spectroscopy (ARPES) and accumulating the signal over many nanocrystals, we find that on clean InAs nanosheets with non-polar surfaces and wurtzite (WZ) crystal structures, a 2DEG can be observed at the Γ-point. We suggest that the step morphology on the WZ InAs specimens facilitates the appearance of the electron gas, since previous studies on InAs nanowire surfaces with the same crystal facet and a similar defect density did not exhibit a 2DEG. Subsequently, bismuth deposition leads to the disappearance of the 2DEG as well as a shift of the valence band. This is in contrast to previous observations on InAs surfaces, in which metal deposition would lead to the formation of a 2DEG. The control of the 2DEG with the addition of Bi atoms is relevant for applications of InAs nanosheets in quantum technologies. This study also illustrates that ARPES accumulated over several 2D materials oriented randomly around their normal axis can provide valuable information on their band structure with a fast turnover and low irradiation. [ABSTRACT FROM AUTHOR]