No, 'House of Saul' equals Apostate Judaism and it's leaders in the days of Christ and the early Church. 'House of David' equals Christ/Apostles/early Church. It's quite an amazing type. Had it all 'roughed out' with highlighter and bookoo marginal notes in my old favorite ASV and drove off to Church one Sunday morning with it on top my car; what a blow, gone forever, have never taken the time to rough it out again. Maybe I should. A.W. Pink makes mention of it as a large type in 'The Life of David', but he does not flesh it out.
House of Saul : Apostate Judaism is rather the same as the corrupted world.
DAvid : Early Church. Good comparison.
However, we must remember that Catholic departed from OT removing the
true trace of the Early Church with the pretext of removing the Judaism.
Jesus clearly declared that He didn't come to abolish the Law, but to fulfill the Law.
He fulfilled the Sacrificial Law at the Cross but didn't abolish the Moral Law like thou shalt not commit adultery.
Today's apostates don't like the Law, LORD's 7 feasts, instead they celebrate Christmas, Easter, even Halloween, but not the Passover or Feast of Tabernacle.
RCC continuously removed the True teachings of OT, blaming they are Judaism. They changed the time and law ( Dan 7:25) as the day of worship was changed from Sabbath to Sunday.
RCC had to remove all the feasts of Bible so that they may bring the pagan festivals.
Eliyahu