This post explains things perfectly.
OP, the emotions of what you've been through this week have made you take this way too personally. To us, pets are family. All pet owners understand this. However, school absolutely cannot issue an unauthorised absence for a sick pet. They have to follow the rules. In reality, there are very few reasons they can authorise an absence for and most of them revolve around personal illness of the child. I've known parents be cross when we haven't been able to authorise an absence because they themselves are ill and can't get the child to school, or a younger sibling was poorly and kept the school aged child up all night and therefore they want to keep them off as they're too tired for school. We can be understanding in those circumstances, in that it's the parents prerogative to keep them off and we can't stop them, but we can't authorise it.
It sounds to me as if you assumed it would be authorised because of your son's impeccable attendance record in the past but it doesn't work like that. Having excellent attendance doesn't give you special privileges to have any old absence authorised. If it's not an authorisable reason, there's nothing the school can do about it. There are certain codes they have to enter to authorise the absence (illness, medical appointment, religious observance, extra curricular activity such as a music exam). If the reason for absence doesn't meet any of that criteria, they can't authorise it.
As I said before, you simply need to let it go as there is really no reason to get so het up about one unauthorised absence. I understand you feel it's blotted his excellent record, but in the grand scheme of things it means nothing. You absolutely won't be in trouble for him having one single unauthorised absence. There really are more important things to worry about.
It does sound to me that school could have been a bit more compassionate in how they handled the situation but the fact remains, they couldn't authorise it. If you reacted to them how you have to some people on here who are are making valid points though, I can understand why any compassion they may have had disappeared. It's not their fault they couldn't authorise it, so you're taking your frustrations out on the wrong people.