After all these decades, I now finally know who the guy with the moon on his forehead is in Inyuasha after only being aware of it for years. The one time I actually watched the series was during a random incident at my cousin's house. Inuyasha was about to start on Toonami, and she wasn't about to give up watching it just because I was over. As years ed, I only knew basic things, like what characters looked like, without knowing their names or even who they were.
This was the case with Sesshomaru. I would see pictures of him and speculate that he was likely one of the stronger villains in Inuyasha but that was it. Now I know that Sesshomaru is Inuyasha's brother, and after reaching episode #9, I can say with great assurance that Inuyasha would likely not have been as popular as it is if Sesshomaru wasn't a character. At the very least, watching Inuyasha would have been more of a struggle for me.
Sesshomaru Was A Welcome Change to the Main Plot
Inuyasha and Kagome's Quest for the Sacred Jewel Shards Could Be Too Repetitive
While I understand that I've only started watching, the effect that Sesshomaru just had on the plot was the first major upset in what's clearly Inuyasha's main premise. It's obvious that the majority of the series will follow Inuyasha and Kagome as they gather the millions of shards from the Sacred Jewel that have scattered across Feudal Japan. While important, just watching them search for these shards and fight against random demons and corruptible humans who want them for their own selfish means would get old very fast.
There are only so many characters, like Yura of the Demon-Hair and the random bandit controlled by Shibugarasu, you can watch get defeated by Inuyasha and Kagome in their efforts to collect the Jewel Shards until things get too repetitive. I could, of course, be wrong. After all, there's a reason why Inuyasha is considered to be the embodiment of turn-of-the-millennium anime.

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When Sesshomaru first appeared, I thought he was just going to be another one of the many random demons looking for the jewel shards, but he offers so much more to the story since he's Inuyasha's brother. In fact, he's not even interested in or cares about the shards, which just adds another dynamic to the story that breaks up its repetitiveness. This formula reminds me of what role the Homoculi seem to play in Fullmetal Alchemist, which is currently on my mind since I just began watching Brotherhood for the first time as well. FMA, to me, needs the Homoculi to break up the monotony of the Elric brothers looking for the Philosopher's Stone.
Inuyasha and Sesshomaru Are More Than Brotherly Rivals
Inuyasha Embellishes These Norms in a Way That Makes Anime Stand Out As a Whole
Regarding Inuyasha, Sesshomaru's relationship with his eponymous brother works so effectively because of how the series capitalizes on their unique bloodlines and powers to embellish the basic mechanics of their relationship. For example, the series gives a fantastical spin to the real-world dynamic of half-siblings, or, worse, racism, and contaminates Sesshomaru and Inuyasha's relationship with them.
This is akin to how half-brothers might have negative feelings towards each other because they aren't fully related by blood.
After establishing that demons are much more powerful and look down on humans, the series inserted that dynamic into this fraught half-sibling relationship by having Sesshomaru, a full-blooded demon, not respect Inuyasha because he's part human. This is akin to how half-brothers might have negative feelings towards each other because they aren't fully related by blood.

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For better and for worse, Inuyasha is the perfect poster child for the anime of the late 90s and early 2000s, when the industry struggled with tech.
While a compelling and ubiquitous trope in fiction, the series takes that another step further by having Sesshomaru mock and manipulate Inuyasha by torturing someone who looks like his human mother. When looking at this from a real-world perspective, a stepchild could easily disapprove of their stepmother for being "the other woman." Inuyasha just portrayed that through how humans and demons view each other in this version of Feudal Japan.
We Have Tessaiga Because of Sesshomaru
Tessaiga Serves as a Metaphor for Inuyasha's Father and Mother
Sesshomaru's character also naturally introduced his father's sword Tessaiga While an awesome blade in and of itself, it serves multiple purposes, including a way to recontextualize Sesshomaru and Inuyasha's rivalry. It was later suggested that the only reason why Inuyasha could unlock Tessaiga's power was because he wanted to protect the human Kagome. The fact that Kagome is human is important because Inuyasha's father, who's also a demon, created the blade to protect his human wife. In other words, Inuyasha's ability to unlock Tessaiga's power serves as another way to illustrate his father's actions to protect his wife.

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Rumiko Takahashi has numerous hit manga works including Inuyasha, Ranma 1/2, and Urusei Yatsura, which set her apart from other mangaka.
Although not stated directly, this could also be attributed to why only Kagome could draw Tessaiga from its resting place, when Inuyasha and Sesshomaru failed. Kagome's achievement further complements her earlier established unique fighting capability in relation to Inuyasha. Since she's a human, Kagome could have easily been overshadowed by Inuyasha, who possesses demonic strength, agility, and resilience. But as the reincarnation of Kikyo, Kagome has unique powers that make her invaluable to Inuyasha. Her skills are so important that he even traveled to her time to retrieve Kagome after she was warped back home.
Tessaiga Also Serves as an Extension to Kagome's Power
Kagome's Ability to Draw the Sword Complements Her Special Sight
Before Tessaiga's introduction, only Kagome could see Yura's hair, and this special ability played a monumental role in helping Inuyasha defeat the demon. After Tessaiga was introduced, Sesshomaru would have most assuredly killed Inuyasha because Inuyasha needed Tessaiga to win but couldn't retrieve it. Inuyasha was only victorious because Kagome could give him Tessaiga.
All of these readings and interpretations would never have been possible if Sesshomaru had not appeared the way he did. Without him, the plot would have most likely just followed the implied premise where Inuyasha and Kagome would have to contend with random demons and humans that sought the power of the Sacred Jewel for their own selfish purposes. Of course, there are ways to make this type of quest more compelling on its own. And while I'm sure that Inuyasha will, the search for the shards no longer feels as daunting as it once did because of Sesshomaru.