A couple of points that come to mind:
* Frozen ATPL is a non-existent licence. It's just shorthand for describing holding a CPL + IR + ATPL exam passes. Some people now include completing an MCC course but historically this wasn't included. Just like FAA & many others it doesn't just magically 'unfreeze' into a *real* ATPL. You must not only achieve the required hours and sub-totals, but also pass a recurrency, licence issue or similar flight test. No different to holding a PPL + CPL theory passes. You wouldn't describe your PPL licence as a 'Frozen CPL' would you?
* Licence privileges of the country of registration are a consideration. Some jurisdictions allow a private pilot to be paid in some private operations eg flying the owner, regardless of the aircraft type. How that would be viewed once in US airspace is also something to consider.