You've disregarded the second, stronger portion of that argument. Constrained though they may be, the trail of precedents judges would generate in the "1000 laws" scenario would simply constitute a complex shadow law.
Common law has tended to focus obsessively on mens rea. Statute law has tended to disregard it. The difference probably has something to do with having to make a judgement about a particular case with the humans that are going to be affected standing there in front of you, as compared to dreaming up a bunch of laws in the abstract, or coming up with laws to address a politically attractive but wildly uncommon situation.
I believe this argument is empirically strong. You can see for yourself that there are (at least) hundreds of years of case law about murder. And yet the law surrounding murder seems quite a bit more straightforward than the laws surrounding, say, exporting electronics. This is because criminal case law has been sharpened by a focus on criminal intent.
http://www.huntingtonnews.net/state/090605-rutherford-statet...
And in any case, arguing around the edges of self-defense is a far cry from the sort of felony paperwork violations that got us into this discussion.
So perhaps the most clarifying question I could ask is: how many examples can you find of case law criminalizing paperwork mistakes with no criminal intent or intent to harm others, as compared to how many examples of that are there in statute law?
Like (I think) most reasonable people, I'm not in favor of criminalizing paperwork mistakes with no criminal intent.