This is a fact: Sesshomaru is traveling a Trail of Power, he wants to gain Power with his own strength and Ways, he finds the Shikon no Tama a cheap way of getting it. He wanted to know Why his Half-brother was Given the Tetsusaiga, instead of Him, and he understood why, so he finds his own way. Here's a little more:
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With the ambition to walk the path of conquest, he desires to become stronger, even at the expense of others. He will not, however, use the shards of the Jewel of the Four Souls to gain strength, which separates him from other yōkai questing for power. Though some think it's for his lust for pure power, others think it is created to accompany his original grace. The Shikon Jewel increases the user's power while swords like the Tessaiga (Iron-Breaking Fang), the Tenseiga, and the Tōkijin attack the opponent using the yōki (yōkai ki) of the person wielding them. Sesshomaru wants to become strong by using his own strength and would never borrow power from another source.