Bidding for "lines" (whatever the hell that is) would stink worse. Days off is about the only thing you can bid for with any small chance of certainty (if done sensibly).
As for the trainers, sorry but they should be flying to all the crap-holes - eg India or Africa so the trainees actually get some benefit out of flying into places where controllers try to kill you and the support is incompetent or virtually non-existent. And for the DECs since they seem to be a necessary evil, they can get some actual idea of what they'll be dealing with when let loose upon the airline after 10 sectors. Not too hard to fly to lovely Europe is it??? Why waste it on training...
Look at the sick days of jr pilots at straight seniority company's as proof. After a while they simply give up and call in sick.
Unlike the old-timers who get shiite flying in their bottom month?? Not really fair either is it? Unless you're junior that is!