I'm a teacher. I think this is a really tricky one. If I found out privately that a colleague abused animals, I would loathe them for it but don't think they would necessarily be a worse teacher. However, if a colleague was publicly seen abusing an animal (and this woman took no care not to be seen which makes you wonder what she'd do when not in public) it would be unacceptable for them to remain in their teaching role.
As teachers we are held to a, somewhat unfair, higher moral standard. One of the lessons we consistently teach in primary school is that violence is not ok, at all. Imagine the playground, little Peter kicks little Thomas and punches him in the face, little Thomas complains, little Peter says 'but Mrs X does it'. I mean how could she maintain any true authority?
Her actions are utterly repugnant, in the role she holds, she should know that her employment would be in jeopardy if the truth about her got out.
What would be interesting to know is if having a conviction for an offence of abusing animals is one which precludes you from becoming a teacher, or contractually if you are convicted, allows your employers to terminate your contract? I don't actually know that. I presume offences of violence against the person are relevant but not sure about animals.
Having said that I do agree that the court of Twitter is a very frightening and dangerous one.