Why have I always got the impression this means either we should somehow try to be weak, or else make ourselves realise that we are weak?

Paul says when I am weak, then I am strong at the end of this passage, not when I am strong then I am weak.  The sequence is this.  First, Paul is struggling.  He is aware of his weakness already and isn’t trying to conjure up some way of making himself feel weak.  Second, he turns to God and God answers that His strength is made perfect in weakness.

Aside: The context here is grace, but the word strength is used also, indicating that this concept applies in other situations where our own strength is insufficient.  Besides, why limit grace to morality?  Anyway I think modern Christian culture in general sees this as applying to more than moral weakness, which is good.

Third, Paul turns around and says when he is weak, he is strong.

This is important to me, because if I believe I am weak, I see myself as less responsible and also less empowered.  If God tells me to trust Him and that I am strong, I am able to be strong in the midst of weakness.  I am able to do the work of love knowing that God is strong when I am weak, and he has made me strong.

There is a huge difference between that interpretation and the debilitating way in which it is often interpreted, a way that makes us afraid to try because it seems unholy to use any strength at all.  I understand that sometimes we need to be shown our own weakness, but that is not what these verses are about.  The context is that of already acknowledged weakness.  It can still be used to give us strength to acknowledge weakness since it tells us that God is there to catch us, but maybe when using it in that context we should be careful to ensure that the person who needs to acknowledge weakness focusses on the strength they can receive in order to accept the fact that they are weak.  The object must always be to grace us with strength.