No, I don't play ME3 whenever I have real life problems on my plate. If it work for you and it helps you, then there's no harm in playing the game whenever things get tough. 
For me, personally, that doesn't work. If I have stuff on my mind, then I should probably want to deal with them straight away. Whenever I feel like crap, I play like crap - basically. In that case, it's best for me to avoid ME3 for a while. If I do play, then it will just lead to more frustration. Working out seems to work better, in that regard.
But to each their own. If playing ME3 relaxes you in those moments where life tests you, then that's great.