My view of the qsearch seems a little different from yours but perhaps we have not the same answer to the basic question: "why do you need a qsearch ?"
Let's take a first basic example.
Take us a branch of the main search in which the following scheme appears for a long time.
You may evaluate here a advantage for white because four black men are blocked by only three white men. But it is black to move and it remains just one move to reach a leaf of the main search.
If black plays 12-17 a static evaluation of the position may now give black an advantage due to white men inactive on the side. 12-17 is probably a bad move but you have to discover the response 21-17 etc.
How can you solve the problem?
You can of course improve your eval function to take care of such basic tactic issue but if you take a slightly more difficult example you will not be able to solve the problem will you?
For me the very basic idea behind the qsearch is to solve this specific issue: detecting a bad last move which changes some bad position in an apparently good position.
Let's take a second basic example
Here it is white to play the last move of the main search and consider the move 30-24.
The situation is very different: the last move 30-24 appears as an agressive attacking move instead of a probably bad move. You have here to define clearly what you want exactly because the situation is quite difficult: with 30-24 white gives black a tempo and it is a typical situation in which a complex combinaison may occur for black, but on the other hand, if no combinaison exists white may win a man.
In order to avoid endless qsearch due to a long serie of unstable positions you may have to accept here a rather bad evaluation. In such situation my choice in the qsearch is only to look for a small combinaison for black in order to prove that 30-24 may be a bad move.
To summarize, the only goal of my qsearch is to try to prove that the last move may be a bad move. I do not look for proving that the last move may be a good move against it no good answer exists.