I never said they were good Jews, only that they and those around them considered them to be Jews.
Even Herod the Great apparently observed at least some of the Law, as Augustus famously observed by Greek pun that it was better to be Herod’s pig than his son, after one of the intrafamily Herodian bloodbaths.
I also believe some of the later Herods, notably Agrippa I, were considered Jews, and even good Jews, by even the strictest. Josephus and the Talmud certainly speak highly of them.
Arippa had a Jewish mother.
It is well-document that the first Herods had an invalid conversion, the second had no conversion at all, and both did nothing more than pretend to keep the law (note your comment about his keeping of pigs — not sure if that was literal, but I bet it was), and neither was born a Jew.