Insurance claim

Not quite. this section refers to vehicles that have never been registered in California namely by a new or non resident of the state.
The only stipulation of mandatory inspection by the CHP is this:

If a New York Salvage Certificate is accepted as proof of ownership, the vehicle must be referred to the CHP for verification of the confidential number.
also:

If verification has been done by the CHP no further verification is necessary.
The Verification of Vehicle (REG 31) form or an additional verification of vehicle is not required when a California Highway Patrol (CHP) Certificate Inspection (CHP 97C) form is submitted on a revived junk or salvage vehicle application.

What you described in your previous post was exactly the process I undertook.