Converting a non-stationary random process into a WSS process by adding a random phase
It is not true that you can convert any non-stationary random process into a WSS random process by just adding a random phase. What is true is that you can make a (wide-sense) cyclostationary process WSS by adding a random phase. A common example is the PAM process mentioned in your question:

$$x(t)=\sum_ka_kp(t-kT)\tag{1}$$

The discrete values $a_k$ in $(1)$ are assumed to be random. The process $x(t)$ can be shown to be cyclostationary. The modified process

$$\tilde{x}(t)=\sum_ka_kp(t-kT+\theta)\tag{2}$$

where $\theta$ is a random variable uniformly distributed on $[0,T]$ is WSS. This random phase reflects our uncertainty about the phase of the signal, i.e., about the origin of the time axis. In this sense, it is physically motivated, and not just a nice "trick" enabling us to compute the power spectrum of $(2)$.