Sometimes it doesn't matter how hard you try because in the end the Mourinho used for Roma was not working and he definitely had a bad performance in this team. Roma did not have enough good players during the last season and Mourinho was unhappy about that but they spent too much money and hired many expensive players to help their team. After that Mourinho was again unhappy about the performance of the team and he also started blaming the referees because of the bad performance they had.
Mourinho is ambitious and professional with his management but his mouth is too uncontrollable and it costs him some suspended matches. It is a minus point from his coach management in Roma and I prefer to agree with anyone call it is unprofessional point from Mourinho.
He asked too much for budget of player transfers and conflicts occur between Jose and Roma's owner and board members about club strategy for development. It is a big condition with Jose to stay with Roma or leave. We all saw what happened, conflict was not solved and he was forced to leave by a sack from Roma.
Now with De Rossi, they are a fresh team as you said and I can see Roma is much more motivated than before. Roma has even a good chance to get the ticket for the Champions League.
De Rossi is a Rome person, he was born there, grew up by Roma Academy and played for Roma in many years. He loves the club as a player and now as a coach and he does not ask too much because he knew the club's limited finance. De Rossi is a coach who will work for Roma, simply with his love and dedication for the club as same as what he did as a player through his career.
With 7 matches and 3 points less than Bologna, Roma has good chance to catch Bologna. Even the fifth position can be enough for them to be eligible for Champions League if Italian clubs perform well in this season from Europa League to Conference League.