Depending on the parameters, the AR, MA and ARMA can be either stationary or non-stationary. For instance for an AR(1) process, if $|\phi|<1$, the process is stationary and else it is non-stationary. But why do we restrict ourselves mainly to stationary processes in the theory of AR, MA and ARMA?
I know that ARIMA can be used for non-stationary processes by differentiating the process until it is reasonably stationary but is it possible to fit directly to our non-stationary time series a non-stationary AR, MA or ARMA model?
From my understanding, it seems like if processes were not stationary, we would have a hard time estimating the mean, variance and autocorrelation of the process because they would change at every time step. This is to contrast to the case where the process is stationary, which implies more parsimony in the parameters to estimate for the same amount of data. Hence, if we estimate a non-stationary model, the quality would be very poor compared to the stationary case.
Are there other reasons why we consider only stationary processes for AR, MA and ARMA models?