Well she didn't have to publicly execute him

Yes she did. She needed to make it clear that she won't tolerate people breaking her laws in her city. The former masters have already seen her wrath, the former slaves needed this.

The freedmen were her leal supporters, arguably the only supporters she has outside of her Unsullied and Second Sons, both paid for in one way or another. As she said herself, they took the City, they've also supported her during her rule & they've bared the brunt of the repercussions for her leadership and it will only get worse as the Son's of the Harpy become more violent against the freedmen.

Aegon V had much the same issue with Brynden Rivers but he instead offered him the Wall, because he knew there would be fierce backlash if he immediately executed him. There are other routes to take to show you mean business, lopping off the head of (what we can assume) the leader of the freedmen is not necessarily the right one to take. Heck, even if she was hellbent on executing him, publically was still the wrong way to go. As Hizdahr said, it should have been done in the Great Pyramid and left there. Instead she did it in the sight of everyone, something which would have ended in bloodshed whether it was the Harpy or Mossador.

She didn't have to kill him. Just take his fingers/hand/genitals.

I don't see many leaders answering murder with anything less than death.

See above. Aegon V did, a fair bit of the Night's Watch are murderers who were spared by their Lord by taking the black. It happens more than you seem to think.

Eitherway we're arguing against a plot point that likely would've happened in some form or another. This was clearly a way for the writers to show Dany's failings as a ruler as they appear in the books, though in a visual format / not inner-dialogue or tiring private conversations.