Why did the Umayyads spare some of the male members of the Family (Ahl-al-Bait) after Karbala?

Question

Given that it's the Islamic month of Muharram, there's something that I have always wondered about the Battle of Karbala fought between Umayyad Empire and the family of the Prophet Muhammad.

In the battle that was fought for annihilating the risk that the family of the Prophet posed to the nascent Umayyad Empire, Umayyad army killed nearly every male member of the family. There were however some exceptions like:

1. Ali Bin Hussayn (eldest living Son of the Imam Hussayn, did not participate in the battle due to sickness). Some sources report that Umayyads did intend to kill him as well but spared him on intercession of his aunt Zainab bint Ali. It seems very uncharacteristic however that the men who would go on to put Prophet Muhammad's eldest grand daughter in chains soon afterwards, humiliate her, respected her so much that they would listen to her at all.
2. Muhammad Al-Baqir (son of Ali Bin Hussayn). He was a child of some 3-4 years at the Battle. The Umayyads had killed his 6 months old uncle Ali Al-Asghar in the battle so it seems unlikely that his age might have been the reason why he was spared.
3. Zaid Bin Hassan (son of Imam Hassan) Spared for unknown reasons.
4. Hassan al-Muthanna (son of Imam Hassan) Fought and wounded, taken captive, spared for unknown reasons. Some sources report one of his maternal uncles was a high ranking commander in the Umayyad army who saved him.

The traditions from different authors are even more confusing due to conflicting accounts and possible sectarian bias.

For the life of me, I can't rationalize why a tyrant like Yazid I would spare some men of his rival family, fully aware how much trouble they could cause him (Umayyad Empire was at one point about to fall during reign of Marwan I due to Hashemite rebellion and even the successor Abbasid Empire had to fight claimants from the Ahl-Al-Bait). So why would he spare them? Mercy? I feel it very uncharacteristic to see some mercy from people who did not even spare infants. Some political motive i.e. pacifying the supporters of Ahl-Al-Bait? It didn't pacify them at all, if anything, it made them hate the Umayyads forever. So why would the Umayyads do that? Certainly it would have been very easy for them to put an end to "Caliph from Ahl-al-Bayt" question for all eternity after the battle?

Answer 1

Yazid went by the standards of the day, but went about them in a somewhat twisted way. Women and children were off-limits for killing, but anyone who took up arms against him was killed. Husayn was holding his six-month-old son in a gesture of peacemaking, but both were killed when he approached Yazid's army. Likewise, while they spared Ali ibn Husayn Zayn al-Abidin because Zaynab bint Ali used herself as a human shield, they carried Zaynab off in chains sans-veil for her insolence.

So yes, Yazid had the interest of removing the threat of Husayn's group, but still (for the most part) held to the tradition of sparing women and children. This meant that other than the six-month-old (who may have been in the line of fire if his father was carrying him towards the opposing army), Yazid held to the letter of the law. Now, did he do so from religious belief, or so that the people of Kufa's tenuous support for Ubayd Allah ibn Ziyad didn't fail? Maybe a mix of both? It's hard to say.