1- They Imams generally didn’t have much freedom to do Aza since there was heavy persecution by the corrupt governments of their time.

2- The hadiths we have indicate they would cry excessively, and one hadiths states their women would wear black.

3- We have sahih hadiths that showing extreme grief and despair for Imam Hussain (a) is recommended, so anything that is considered as such becomes mustahab as long as it’s not haram (hitting the chest, etc…). Scholars who say Tatbir is mustahab generally based their fatwa on such hadiths.