Joe,
In the court of world opinion, Britain would have gained much, if it had only refrained from executing these people. Instead, it created martyrs and lost its case completely.

I agree.
It brought the Irish people to finally side with the rebels.

Perhaps the British should have foreseen that and politicians intervened sooner to stop it, but consider the context.
Britain, including Ireland was in a terrible war and suffering casualties on an historically unprecedented scale.
Irish volunteers included.

Not an excuse bout an explanation.
What explanation is there for the executions of the civil war?
In what sense do the rebels have the moral high ground?

why it would be proper for the English to have a presence in Ireland.

Why is it proper for Scotland and France to be part of the UK?
It has just come down through history in that way, and will change as soon as the people of Wales and Scotland want it to.