A couple of minor points:
You can act as PIC with only a CPL while doing commercial air transport. It just can't be in an a/c type that requires more than 1 pilot. Quite a bit of commercial air transport is done in single pilot types. I suspect you're considering only airline type ops that use multi-pilot aircraft. They're the ones most people think of when they're referring to air transport ops.
A US ATP includes an integral IR. An old UK, JAR or many other countrie's ATPL doesn't so you must also hold an IR added to the underlying licence.
It's quite true that the most common route is PPL to CPL to ATPL. Depending on the country an IR could be added anywhere after the PPL (except the US where you could front up for an ATP test without an IR).
It's also possible in some countries to go directly to a CPL without ever holding a PPL eg in Australia.