No, the barons mental health did not play a large role in the loss of Arrakis.

The loss can almost entirely be attributed to the mercy shown by the traitor Yueh in allowing the escape of Paul Atriedies. If there was no Muad Dib then the Harkonen's would mostly likely still control Arrakis.

While the Baron has his eccentricities, it is worth remembering that we was still a very shrewd individual. His plot involving a traitor in the Atredies household came very close to ending the house entirely.