The team is picked based on perfomance in training and if he isnt trying his hardest or isnt impressing then why would you start him? He isnt even that great, yes he is intelligent with runs and scores the odd goal but he hasnt made one successful dribble or cross all season (according to Opta)... And he plays on the wing.
He has scored 3 goals in 10 appearances. Last season his goal record under Dalglish was superb. He has something like 11 goals in his last 14 starts hasn't he? Now given we lack goals, I'd say that is fairly important especially when we compared it to Downing's 0 goals in the league (and 0 assists for that matter - although that is not totally his fault, he can't finish the chances he creates.)
Players who play wide don't just cross, some like to cut inside and play one / two's something Suarez thrives off and they look great together.
I saw an interview where Souness and Dalglish met Torres, Reina and Arbeloa. Souness said (and Dalglish agreed) that Dalglish was the worst trainer at the club when they played, but it didn't matter because on a Saturday he was the best player and did the business. Now if Maxi is doing the business when played, who cares if he isn't the best trainer?
Also I find it hard to imagine he isn't a good trainer. Dalglish has always praised his professionalism and attitude, plus with his technique he would struggle to look poor in training.
I might be wrong, I just think Maxi doesn't play because FSG want rid of him due to his wages / age and Dalglish wants to build a team to play together for 2, 3+ years and plays Henderson, Downing etc over him as a result.