Once upon a time it was easy:
Arrive in the US with an ICAO CPL or ATPL licence, do the theory exam, go to any examiner & do the flight test. No training (but recommended. Things are a bit different from Oz) or instructor sign off required.
Then insecurity paranoia got a foot hold and the TSA waded in with their useless contribution to 'security'. It added a requirement for a background check with fingerprints prior to training. That, in turn, requires a sponsoring school so rather fewer schools are approved for that.
Then the 'we don't trust your licence you presented to us' so now you had to add the back & forth 'release my info to the other regulatory body' stuff.
It was still easily do-able, albeit with more paperwork & time to prepare required.
Then a year ago the US changed their prerequistes for sitting the ATP exam (for multis). Now you have to do an approved preparatory course that includes time in a full motion sim. Very, very few organisations are approved for this.
So...it's no longer the quick, easy & cheap ATP it used to be!