What is the rationale for not allowing jurors to see a written copy of verbal directions they were just given? What is the rationale for jurors to not have a written copy of the laws cited by the judge as applicable in the case?

You're absolutely right. The rationale is that you don't want the jurors getting involved in debating the law, or debating the rules that they're supposed to deliberate under, on their own in the jury room. If they have a legal question, they're supposed to ask the judge - so that the judge can answer it, in open court with both counsel present.

It's not a great rationale. Several states (including Florida) go the complete opposite way - they require written jury instructions. That way there's no misremembering or misstating what the instructions are in the jury room.

But I do understand the general principle, particularly as to statutory text. Sometimes statutes are written in ways that are confusing or admit of different interpretations, and sometimes those interpretations have already been ruled on. So if a statute appears to say "X" instead of "Y," but the state Supreme Court has ruled that actually "Y" is the correct interpretation, you don't want the jurors reading the original statute and forming their own - not the legally correct - conclusions.

While I personally think the "written instructions" rule is the better one, the law is the law. Different states have different rules. So when some blowhard gets up and proclaims that they've been done trials for XX years and they've never seen this happen, but they don't practice in NY, you should take it with a grain of salt when they claim that means the judge has made a mistake or is violating the rules.