Like others have said, C-Kira was NOT Kira.
C-Kira was someone who simply obtained the Death Note and decided that it was best to use it on the elderly who begged for death, and as seen later with his escalation, ANYONE who begged for it. Way different MO than Kira, who actively targeted criminals and those he deemed were obstructing his justice.
But that means C-Kira was just a mercy killing serial killer. No grand motive, no grand ideals. And when they were called out, they folded hard.
It could have been their compassion backfiring.
‘I’m doing this for their own good, I’m doing the country a service - OH GOD I KILLED ALL THESE PEOPLE’
It could have been their ego being shattered.
‘L doesn’t see me as a WORTHY opponent?!?’
‘No, I AM Kira! Don’t you see? Is this not what Kira would have done?’
No matter the justifications, the excuses, and what the reasons were, the cold hard truth was that their actions were plain simple murder.
So they offed themselves.
In a meta perspective, their story mirrored Near’s: whereas Near successfully became his own version of L, C-Kira couldn’t become Kira, and couldn’t accept the version of Kira they were becoming.