God hates divorce, plain and simple.God hates sin is how I would put it.
Nevertheless, where in Jesus' statement do you find his approval to remarry?
Here: "But I say to you that whoever divorces his wife for any reason except sexual immorality causes her to commit adultery..."
At that time, a Jew could remarry after any divorce. Jesus has just narrowed the permissibility. The implication clearly is that the sin of adultery can destroy a marriage. I don't believe that Jesus expects someone to remain in a marriage when the other party treats the institution as a farce.
Separating because of an adulterous spouse is because forgiveness is not forthcoming from the abused spouse. Even so, if they put the offending party away (divorce) given the permission you note, you still miss the other question, namely: Where is the Lord's permission/blessing to remarry someone other than his/her original spouse? Point being ... HE NEVER GAVE APPROVAL, PERMISSION OR BLESSING. Why? BECAUSE IT WOULD BE ADULTERY! If the couple reconcile or if one spouse dies. Those are the only allowances provided. All else is not blessed by Him.