More like he wants to keep it the strongest one around. [And not (just) for crazy shounen powerlevel reasons; he has subsidiary houses to keep safe, with the fates of hundreds of people riding solely on his shoulders.] I don't know if it's the same where you come from, but many exorcist houses are on the decline, with their spiritual abilities growing weaker or in some cases losing the ability to see ayakashi at all. Most of them choose to flock to the Matoba clan for protection, so the Matoba are responsible for their safety as well.
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