Why does the Duke so readily dismisses Brabantio's suit against Othello in Othello?
This is a very important scene in Act I scene 3, but let us remember the context, which will help us to understand the answer to the question. The scene begins with the Duke discussing the military manoeuvres of the Turks and how they are threatening Venice. Even before Othello and Brabantio enter and they know about Brabantio's suit, they have already decided that this is a matter of some urgency and that they must use Othello to counter it. Note the way the Duke, focused on the task at hand, immediately greets Othello as he enters:
Valiant Othello, we must straight employ you
Against the general enemy Ottoman.
It is clear therefore that the Duke is keen to dismiss any issues that might prevent Othello from starting to do his job. However, what really shows that Branatio's charge against Othello is false is when his own daughter, who is now Othello's wife, comes to speak and testify on Othello's behalf. Desdemona's words are enough to convince even Brabantio that Othello did not use foul means to steal away his daughter. Note how he responds to Desdemona's speech:
God be with you, I have done.
Please it your grace, on to the state affairs;
I had rather to adopt a child than get it.
So, as much as the Duke is keen to focus on the affairs at hand and wants to sort out this matter as quickly as possible, at the same time, Brabantio is so convinced by his daughter's words that he himself relinquishes his suit against Othello and tells the Duke that he may return to state affairs.