Ah, but even in the civil law system, there was a similar period of judge-made law.
There were the 12 Tables, which apparently set out some rules of Roman law, but there wasn’t much by way of statutes. Instead, there were customary practices, which started to become the equivalent of customary law.
Gradually, there were individuals who were skilled in the customary Roman law, who gave their opinions, sometimes individually, sometimes as a group, to individuals who had a dispute.
Over a few centuries, those opinions became the basis for Roman law. They eventually were written down on the direction of Justinian, and that complex of written laws, customary laws and opinions became the basis for Roman law studied in the Italian universities at the beginning of the Renaissance. Padua in particular was one of the most important universities studying the re-discovered Roman law.
And statutory law didn’t happen overnight. For example, in France, there were a number of different regional bodies of customary law. The “coûtume de Paris” was particularly influential (it became the basis for Quebec’s variant of civil law), but the legal principles varied depending on where you went in France.
That all got swept away by the Revolution. The idea of France as a single nation took hold, and unification of the law was a natural result, just like the metric system replaced all the regional systems of weights and measures.
That’s the origin of the Civil Code, compiled on the direction of Napoleon.
Another key part of the Enlightenment was the idea that criminal law should be written down and defined by the Legislature, not left to judges and case law. That resulted in the French penal code, replacing all the judge-made and customary offences.
The French legal innovations in turn became part of the revolutionary ideology which the French exported to continental Europe, so written civil codes and penal codes became part of the European civil law tradition.
Note that England never joined in that revolutionary impulse, and thus still has judge-made law and common law offences.