Maybe we have a difference in definitions here. I think Mr Bennet has those faults in character you discuss, such as letting Lydia go to Brighton, or teasing his wife and daughters. I wouldn't call those faults wishy-washyness. By my mental picture of wishy-washy, a wishy-washy Mr Bennet would have been planning a family trip to Brighton at his wife's bequest, until Jane and Elizabeth sit him down and convinced him it would be too expensive, then Mrs Bennet would get hold of him again and he'd go back to planning the trip, then Jane and Elizabeth would intervene, until the invitation came from Colonel Forester, and he'd approve it, then Elizabeth would have talked him out of it, then Lydia would have talked him into it, and back and forth until the reader got quite dizzy. Harriet in _Emma_ meets my definition of wishy-washy (all further discussion to Austenations).
We don't know how long Jane and Elizabeth spent persuading Mr Bennet to invite Lydia and Wickham to Longbourne, so I don't have an opinion about whether he was quick to give in. Though given Lydia's behaviour, I don't think anything could have convinced her that she did any wrong.