It can take several attempts to leave an abusive relationship. It's more than frustrating for an outsider looking in to observe and watch. It must be heartbreaking for you?
Staying together for the kids is a complex topic and it's more difficult because of high societal pressure as much as many would say doesn't exist, it's there and insidious for any person considering leaving.
I agree that it's not true. Children living with alcoholism/addiction and a presumed personality disorder which I mention because of the abuse, aren't left unaffected. They might not be on the receiving end of the blows, but life in a hostile home cannot be healthy. Gosh, my teen, recently needed to note a key figure that was affected by their parent for a homework assignment. My son gravitated right to a celebrity whose father was also an alcoholic. I gasped to realize how much he identified with this famous person. My son was 7 when I divorced his father.

There is the burden of finances in leaving. There is the burden of the two parents at all cost societal pressure. There's logistics and more.

And as someone else pointed out, removing the alcohol alone might not resolve things. My ex is a miserable dry drunk, to be honest.