Does this whole thing depend on countries not publishing their registered companies online? And becomes immediately transparent when they do?
In depends on the fact that if US investigators come calling, all the government will tell them is "fuck off."
However, the real issue is that very powerful people are using these techniques. So, governments have to worry that they're going to accidentally trip over somebody who is powerful enough to cause problems. If you are a bureaucrat examining what you think are low-level offenders but accidentally stumble onto something held by Vladimir Putin, you're going to bury that post haste so as not to endanger your life.
A government is not going to threaten to murder folks for not turning over some financials. I think you've been reading too much Tom Clancy.
E.g. I set up a trust and appoint a law firm to handle my affairs (I can already hide behind client attorney confidentiality). My lawyer sets up an offshore company and appoints nominee officers. That company sets up another offshore corporation in another jurisdiction again with nominee officers. Etc.
To get back to me you would have to find company information, find who those nominee officers represent (which would require international warrants) and follow the trail. Rinse and repeat until you eventually get to my lawyer who still has few ways of preventing you finding my name.
And you can do many more layers of hiding than the three illustrated here
I register a company in small country A.
As a representative of that company, I open up another company in another country, preferably one that will not extradite and has a history of a lack of cooperation in criminal and civil cases.
They then use that company to open up another company in Seychelles which in and of itself according to this post is a known place to siphon funds through a corporation.