As an owner, for new home construction, does filing a Notice of Completion allow sub-contractors who finished long ago to lien?

This response does not create an attorney/client relationship nor does it constitute legal advise but is provided only as a point for discussion.
It is my understanding that a properly filed Notice of Completion is the point from which time to file a mechanic's liens is calculated. For a subcontractor, it is always better to file a lien early rather than too late meaning that liens are often filed based upon the completion of the subcontractor's work.
If you don't file a Notice of Completion or you file a defective Notice of Completion, subcontractors have 90 days from the "completion" of the work to record a lien. If you file a proper Notice of Completion, it cuts down the time to file a lien from 90 days to 30 days after the notice of completion or cessation is recorded.
There are other requirements for filing mechanic's liens that must be met depending on whether a contractor is in direct contract with the owner or is a subcontractor of another subcontractor. Defective liens are invalid. Materialmen have slightly different rules as well.
CA Civil Code covers these areas. You can access the code sections online and also the the California State Website has a large section dealing with contractors with information for consumers at cslb.ca.gov.