The word has multiple meanings, two of which you have covered with your translations.
Most of the connotations this word implies are negative; in fact, only the fish-sentence can be understood in a positive way. I’m pretty sure that most of the meanings derived from somehow sitting something out or making something sit outside; it is now a rather widespread verb that can carry dozens of subtle meanings in context.
Note that etwas an etwas aussetzen usually requires a complement with the particle an — not present in the OP’s quote. However, I have heard the occasional German omit that complement if both parties were absolutely sure about the object that was being criticised. Omitting daran is not formally correct, though.