Do today's professional astronomers use earth rotation angle or sidereal time? I have read that "earth rotation angle" is the modern replacement for "Greenwich sidereal time". Is it actually the case that modern professional astronomers are using earth rotation angle? Either way, what is the reason for using the one they use?
Most professional astronomers tend to use whatever package they are used to or got from somewhere else and don't look "under the hood". Very few astronomers I've met tend to write the own code that needs to deal with the multiple time and co-ordinate systems needed.

Most tend to use AstroPy these days which hides a lot of details but uses the SOFA/ERFA routines underneath which do indeed use the modern IAU2006/2000 framework and Earth Rotation Angle. However most textbooks, webpages etc still use the old methods to compute an apparent place and then sidereal time to rotate and relate that to the terrestrial system. So a lot of people still use the old methods because that's what they learnt initially or read recently. Until there is an update of a readable book like Jean Meeus's Astronomical Algorithms to the new system, I suspect sidereal time etc will persist for a good while longer.

The two methods give the same answers for positions to more precision than most astronomers can measure for most objects; exceptions would be pulsars, quasars, spacecraft (all three mostly done in the radio and/or with VLBI techniques) and GAIA-related measurements.