Thus whatever Mu'awiyah did...
Thus whatever Mu'awiyah did, if corrigible and justifiable at all, it was rectified and justified by these authorities; otherwise, by giving a great deal of hush-money, he prevented his opponents from complaining. Where these means and methods did not work, thousands of innocent people comprising of Shi'ahs (friends) of 'Ali (AS), other Muslims, and even a great number of the of the Holy Prophet (SA) were killed by these collaborators, partisans, and "companions" of Mu'awiyah.
Mu'awiyah pretended to be right in whatever he did and accomplished everything with a special patience and forbearance. With a particular gentleness and flexibility, he attracted the kindness and obedience of people. Sometimes, he even heard the curse of people on him and sensed their enmity, but he responded with cheerfulness and forgiveness and thus pursued his policy in this way.
Apparently, he paid respect to al-'Imam al-Hasan (AS) and al-'Imam al-Husayn (AS) and sent them valuable gifts and souvenirs. On the other hand, he publicly announced that whoever narrates a hadith concerning the virtues of the Ahl al- Bayt (AS), will have no security of life, reputation, and property; but whoever narrates a hadith concerning the virtues. Prophet (SA) will receive a of the of the Holy reward.
He gave orders to the orators and preachers to curse al-'Imam 'Ali (AS) from pulpits used for addressing Muslims. By the order of Mu'awiyah, his hirelings killed the supporters of al-'Imam 'Ali (AS) wherever they could find them. They went along in this matter to such an extent that they killed a large number of people who were the enemies of al-'Imam 'Ali (AS), accusing them of having friendship with the Imam (AS).
It becomes clear from the above description that the rise of al-'Imam al-Hasan (AS) would only have ended to the detriment of Islam and would have had no effect other than the martyrdom of the Imam (AS) and his supporters.
And even it was not unlikely that, in this case, Mu'awiyah might have got killed al-'Imam al- Hasan (AS) through the friends and relatives of the Imam (AS) and then, in order to pacify public opinion, Mu'awivah would have rent (torn) his shirt and mourned for the Imam (AS) and, in seeking vengeance (for this bloodshed), he would have tried to take his revenge by killing the Shi'ahs, as he later did while dealing with the case of 'Uthman.
But the political trend of Yazid had no resemblance with that of his father. He was a self-admiring and indisciplined youth.