Was I right in my previous analysis, that composer's rights and arranger's rights have to be totally independent of the rights to the sound recording? Because there may be third parties involved, who have no contract with the recording company. (i.e once a song is published, it can be recorded by anyone)

If the CDR does not carry the MCPS logo (either on the CD or the insert) or equivalent for the country of production, then there is no guarantee that this legal requirement being observed.

If the CDR DOES carry this logo, then the composer and arranger should be getting appropriate royalties from the MCPS, which would be easy for the owner of the rights to check.

So do they or not? it should be simple to check.