I've recently read up about time standards and now understand that UTC is a second-corrected version of international atomic time so that it is kept within 0.9s from UT1. And then UT1 is defined as the mean solar time at 0° longitude. However, it is not realized by measurements of the sun's apparent position, but via a sidereal time or earth rotation angle.
Extensive googling hasn't helped me to understand how this observed earth rotation angle is used to compute UT1. This Phys.SE answer hints that the equation is quite complex. I'd like to understand why it needs to be complex, considering that the "mean sun" that mean solar time refers to is defined to follow a uniform movement (with respect to the stars) along the celestial equator. Therefore, shouldn't the difference between sidereal time and mean solar time be simply a linear function of the time of year?