It's also faction specific as well. I'm also a Shadowlands resident, and I find I generally have to wait longer as a tank pub side versus as a healer imp side. My imp side healer has insta-queues for the most part and my pub tank has a wait of 0-5 minutes generally.
This is true. So in summation:
1. Tanks OR Healer is always the right answer =) No matter what, outside of a miracle, Healers and Tanks are always more sought after than DPS role classes.
2. It depends on BOTH your server, AND the faction. I have toons on both sides, and sometimes the Tank is insta-group, other times it's the healer - but depends on the time of day, and the side.
3. In general - Tanks are more sought after, imho, followed by healers, and very distantly by dps role classes. I have 3 tanks, 1 healer, and 1 dps at level 55. My dps toon, I mean, I gotta really be in the mood for waiting for a group, if I want to do a group with him. =/ The others are all relatively quick - by the time I check my mail and/or do some housekeeping in my bank or companion management, I am in. With tank, usually I have to not queue, if I want to do anything else =)