It does, to some extent, but it's still much closer to natural language than code. Expressing the law in code would turn this dial up to 11 and then some.

Sure, but there would be all manner of services that could automatically translate law-in-code to any human language you wanted. The best part is that you could easily and automatically translate a law into, say, English, Japanese, and German. Whenever the law-in-code source changes, just rerun your translator and voila: no human intervention required (meaning faster and more accurate translations of the law into human-readable language).

You could even program the "law-in-code-to-humanspeak" translator to generate different levels of the target languages, e.g. translate into something at a 6th grade reading level vs. something at a grad school level. Again, the advantage would be the automaticity.

But now instead of reading the laws that govern my life myself and understanding them, I have to rely on some third party to explain to me what laws my representatives are voting on.

The government who writes the coded law could be required to provide freely available services that translates those laws into plain language.