Phase discontinuities result in spectral leakage. This is because, at any point where there otherwise would've been energy at frequency $\omega_{0}$, there is now energy at some other frequency $\omega$. The spectral leakage doesn't necessarily need to be concentrated at high frequencies, but the spectrum will be spread out.
In terms of power spectral densities, if there is leakage in the spectrum, there is leakage in the PSD, short of using some fancy tricks that require a priori knowledge of the signal or just dumb luck. There are two explanations I see for this, with each one being related to one of the two definitions of the PSD. The first definition of the PSD is
\begin{equation} \phi(\omega) = \lim_{N\rightarrow\infty}E\left[ \frac{1}{N}\left|\sum_{t=0}^{N}y(t)e^{-j\omega t}\right|^{2} \right] \end{equation}
As we can see, the PSD is related to the spectrum through a magnitude squared operation. So, if there is spectral leakage in the spectrum, there will be leakage in the PSD.
The second definition of the PSD is
\begin{equation} \phi(\omega) = \sum_{k=-\infty}^{\infty}r(k)e^{-j\omega k} \end{equation}
According to this definition, to understand why spectral leakage occurs in the PSDs when phase discontinuities are present in the data, we have to understand how phase discontinuities affect the autocorrelation function. Sinusoids have an autocorrelation function which is a triangularly windowed sinusoid with the same frequency of the sinusoid in question. Phase discontinuities in the data make the waveform more unique, which disrupts the envelope of the autocorrelation function (this is actually the basics of phase coded waveforms, e.g. Barker codes). Specifically, it makes the autocorrelation function more of a delta shape. Using the relationship that a waveform which has little duration in the data domain has a broad Fourier response, if we take the Fourier transform of a narrow autocorrelation function, we get a broad PSD. And since phase discontinuities "tighten" the autocorrelation function, we can expect that phase discontinuities will lead to spectral leakage in the PSD estimate.
The reason you aren't noticing the change in peaks as much is probably because of the amount of data you are using. The more data you have, the more the discontinuities get averaged out, especially in Welch's method. In Welch's method, the windowing operation helps to decorrelate the data from frame to frame, to help lower the variance. See this answer for how different phase discontinuities alter the spectrum in different ways. Each frame of Welch's method will have different discontinuities and therefore different spectral leakage, which will be mitigated in the end result by the frame-to-frame decorrelation and averaging.
