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There's nothing separate about this.

Chained just means they used php2shell to run a local privilege escalation.

Was the exploit not dependent on a kernel exploit that was disclosed and patched in may, or am I just hallucinating?

Depends which part: php2shell does what it says on the tin: lets you run arbitrary commands as www-data.

If you want more than that you need to chain an LPE. Which are plenty these days sadly.