There is a very high rate of survival of Sengoku clans into the modern era; even defeated clans have surviving branches, and for Edo Period families, an almost universal survival. This includes the Yagyu. You should check out this question: Do any of the major clans/families of the Sengoku era still exist in Japan? Are there families who still have the Date/Takeda/Oda names and can trace their lineage to the major clans or did they all die out during the Tokugawa shogunate? for a little bit more information about surviving clans. I recommend the book Above the Clouds in that link for an exploration of this topic.

One reason for the stability of Japanese surnames over centuries is the practice of adoption. Edo Period lines didn’t die out in the same way European ones did. Family names could be transferred to adopted heirs, both children and adults. Inheritance could pass through a daughter, but in that case, the daughter’s husband was adopted as the family’s heir, and he took her family name. The actual genealogy of the important Edo period families is a huge tangled network of adoptions among families: spare heirs, both male and female, being shuffled around, loaned out, and returned under the direction of the shogunate which kept control of the overall marriage alliance system.

The importance of keeping a notable family line going wasn’t just personal; the livelihood of a lord’s retainers, and the smooth government of his domain, depended on a transition of headship. On the odd and extreme occasion that a house was disbanded, as in the famous story of the 47 ronin, the retainers became ronin. So, barring exceptional circumstances, the shogunate supported transitions by any means necessary. In his book Performing the Great Peace: Political Space and Open Secrets in Tokugawa Japan, Luke Roberts delves deep into the records to highlight cases where the shogunate turned a blind eye to lords “adopting” heirs after they were already dead, heads of houses being two separate people (with one kid from another family being substituted for another after the first died), and other ruses to get around the laws for properly transferring headship. This sort of thing went on among Tokugawa retainers and other samurai, as well with daimyo. I don’t have the Yagyu genealogy at hand, but they would have had access to all these means to ensure their survival as they continued to serve as instructors in swordsmanship.

The Yagyu are still alive today, although only their Owari branch still have a headmaster who’s teaching their sword style: Yagyu Koichi. The Owari school of the family, which is the main line, is descended from the nephew of the famous Yagyu Munenori. Munenori, the father of Jubei, founded the more legendarily prominent Edo school. Munenori’s descendants bearing the Yagyu name still exist but are not swordsmen. The last Edo line Yagyu to teach the family style was Yagyu Toshihisa. I don’t know his exact dates but he’s listed as a Viscount in the 1920s Japan Year Books.

Changes in name form certainly happened in Japanese history, but they played out very differently because of the writing system. No matter how the pronunciation evolves, the characters for the name usually remain the same. So for example, the rulers of Tosa in the Edo Period have post-Meiji Restoration often been referred to as the Yamanouchi, which is the modern reading of their name, based off standardized Japanese. But the contemporary Tosa pronunciation was Yamauchi. Historian Marius Jansen made the decision to use Yamauchi and other contemporary name readings in his 1961 book, Sakamoto Ryoma and the Meiji Restoration , and the field since then seems to have mostly followed his lead. But there are still numerous English books using the early 20th century Yamanouchi. Through all this, the kanji themselves: 山内 remain unchanged. Kanji make tracking names easier, but figuring out contemporary pronunciations, particularly for given names, a challenge.