Does a phase rotation introduce a delay?
I would assume a discrete-time context (though your notation suggests continuous-time).

Then you have a misunderstanding of the relation between the "delay" and "phase". Delay is measured in time-domain whereas the phase rotation is defined in frequency-domain.

So if there was a delay $\pm d$ between $x_1[n]$ and $x_2[n]$, then $y_1[n]$ (defined as the inverse DTFT of $Y_1(\omega)$ defined below) will have zero delay wrt $x_2[n]$ in time:

$$ Y_1(\omega) = e^{ \mp j \omega d } X_1(\omega)$$