The quality slider is actually quite important and I do not think it should be hidden.
Swap quality and optimize drawing controls
Adding @Sergey Sharybin (sergey) as reviewer. As I see it, Quality isn't really that important since the default value works in most common cases. It's an advanced setting which can fix some corner case issues. All the things in the Advanced panel are things that most users will probably not have to change for most common cases - that's why they are in there.
The definition of "most common cases" is illformed. Current setting works fine for most organic characters, although we do have some occasional reports related on default value not being high enough for some characters. The setting is NOT ideal at all for mechanical objects (robots, props etc).
All those settings heavily depend on what exact scene you work on, what is on it, what part of scene you're editing. (Same applies for creases and UV smoothing, it all depends).
I do not see any benefit of trying to hide some settings of any modifier: you add modifier, you change settings, you collapse modifier. If you want to re-iterate on the settings, you open modifier and should be able to see all the settings on a glance. This division on Advanced and Common is absolutely arbitrary and is always in a way.