Being a US citizen pretty much negates any of the true advantages. We have a global tax system, which means income earned anywhere is taxed by the US, with certain exclusions. The only way to game the US tax code in the way you're describing would be to move to a place that has no income tax, and then earn enough locally to max out the foreign earned income exclusion, standard/schedule A deductions, and exemptions. The countries without income tax are either really poor, so that's hard to do, or have extremely HCOL (Cayman Islands), so you're not getting much for your high earnings.
On a basic level, what you're getting at is basic arbitrage. Earn high US salaries in low cost areas. This might mean living in Croatia and working remotely at a high tech company in San Francisco. This isn't fundamentally much different from doing the same thing from Biloxi, Mississippi. Or, as you mention, get that tech job and live in Australia because the currency is low. In theory, you could move around and follow the misfortunes of the countries where you want to go as their economies crash, but you're fundamentally going to rely on a HCOL US job to arbitrage.
One thing that I find interesting is keeping US dollars overseas. I know a retired guy who lives in Costa Rica, and gets 8% interest on a money market account in local banks. The account is dollar denominated, so there is no currency conversion risk. There is solvency (and other) risk; there's no FDIC to back up that account. Are there options like this in other countries?
