This is from the link you posted.
"His name was Yazeed ibn Mu’aawiyah ibn Abi Sufyaan ibn Harb ibn Umayaah al-Umawi al-Dimashqi.
Al-Dhahabi said: he was the commander of that army during the campaign against Constantinople, among which were people such as Abu Ayyoob al-Ansaari. Yazeed was appointed by his father as his heir, so he took power after his father died in Rajab 60 AH at the age of thirty-three, but his reign lasted for less than four years.
Yazeed is one of those whom we neither curse nor love. There are others like him among the khaleefahs of the two states (Umawi/Umayyad and ‘Abbaasi/Abbasid) and the governors of various regions, indeed there were some among them who were worse than him. But the issue in the case of Yazeed is that he was came to power forty-nine years after the death of the Prophet SAWS (peace and blessings of Allaah be upon him); it was still close to the time of the Prophet and some of the Sahaabah were still alive such as Ibn ‘Umar who was more entitled to the position than him or his father or his grandfather.
His reign began with the killing of the martyr al-Husayn and it ended with the battle of al-Harrah, so the people hated him and he was not blessed with a long life. There were many revolts against him after al-Husayn, such as the people of Madeenah who revolted for the sake of Allaah, and Ibn al-Zubayr."
Yazeed wasn't appointed he was elected. The problem with this belief is that it portrays that Shabah had a weak character, because it says that they didn't object to Kingship, which is impossible. Plus Hazrat Yazeed's father was reluctant when Shabah told him to appoint a heir to Caliphate and insisted that it should be Yazeed. Yazeed's father didn't wish that, so a grand referendum was held where common people got to vote, the first ever biggest democratic selection.
Why Shabah recommended Yazeed to his father? It is already given the above mentioned account, if Yazeed was worthy of leading Muslim armies into a battlefield, the army which included people like Abu Ayub ....... then he must have had a solid character of a true Muslim, he must have had qualities worthy of leading.
As in regards to Yazeed killing Hussain, why would he wait for Hussain to reach Karbala, if Yazeed was so adamant to kill him, and given that he wasn't that good person than he would kill him anywhere well before Karbala. One thing
Second thing .......... Karbala is ahead of Kufa (the intended destination of Hussain), if Hussain intended to go to Kufa then what was he doing in Karbala?
Third thing .......... there were Hashmite women married to umayyads, close relatives of Hussain married to close relatives of Yazeed .... if Yazeed knowingly and intentionally killed Hussain ....... then why would those women remain in Nikkah of their enemies?
Fourth thing ........ Yazeed gave the best irrigation system. So he wasn't drunk incompetent loose character.