I cannot understand why this doesn't introduce a set of unwanted high-frequencies, as the steep changes are supposed to reflect as so in the frequency domain.
The steps in the time domain for unshaped PSK refer to a rectangular baseband modulation waveform. The frequency response of this is a Sinc function which does indeed contain high frequency components, compared to a pulse shaped version where the time domain transitions are tapered rather than stepped.
When modulated to a carrier, the spectrum at baseband is simply frequency translated to the carrier frequency. Thus at a carrier the spectrum associated with the unshaped BPSK waveform will have wider occupied bandwidth, due to those higher frequency components.
It is typical to shape the PSK waveform such as with root-raised cosine pulse shaping, to limit the occupied bandwidth and therefore improve the spectral efficiency. The resulting waveform the OP is seeing is not shaped. To see the significance of this, I added a plot showing the frequency spectrum for PSK waveform without pulse-shaping (blue) and with pulse-shaping (red). The step in phase shift introduces significant high frequency components that can be significantly reduced if stepping is not allowed.
And here is a portion of the modulation waveform in the time domain with and without pulse shaping. It's rather straightforward: A slow change in time corresponds to lower frequencies, a rapid change in time corresponds to high frequencies: