Talking about investigation responsibility and costs is a diversion to this thread.
This properly resides in ICAO annex 13, but since those are recommendations and not law, some variations occur (mostly revolving around deep pockets vs holes in pockets)
Argiuments can exist for the manufacturer having the responsibility for correcting known airworthiness issues of aircraft in service. Since crashed aircraft don't belong to the manufacturer and are no longer in service ....
OK, the manufacturer has a need to respond to known deficencies as reported by the operator or investigating agency (BEA) to the satisfaction of the regulator (DGAC).
The manufacturer typically carries insurance for unexpected costs (deep ocean recovery is excluded)
ergo the responsibilty rests almost entirely with the investigating agency to make investigating decisions which can be economically supported by themselves.
No harm in anybody else offering support for the investigation (persons or money) but the buck stops there.