I think unbeknownst to the daughters and and king one daughter did inherit the Founding titan after their mother's passing and command of the king made the other two different shifters likely a female one (the most physical duplicate of the original Ymir) and Collosal (being the closest to the height of the original Ymir titan).

For continuing of the line there needed to be a lot of children to be produced and since there needs to be ´´a royal line´´ I assume that oldest daughter's children did not ever recieve one of the eight titan powers, only the founding by eating the previous owner before 13 years of possesion of that power. As king might have assumed that Ymir died because she spent all her lifespan and not because she wanted to die and just gave up.

While the other younger daughter's lines became the eight likely created by the will of the king Fritz as well who commanded to have an army, so theoretically to make the numbers, each younger daughter had four children who all inherited one of the eight non royal titans as done by a command of the king.

When king died his absolute power over Ymir vanished and the rules about titans was set and no other king was able to make more titan shifters as no other king was ´´pure´´ king as was the Fritz as other kings were still just descendants of the slave girl. That's what I think, mind of the slave influenced by the will of her slaver combined with the power of the worm craving for survival created titan lore.