SA to FAA ATP
I arrived in the USA 2 years back with an SA ATP and got my FAA ATP in 2 weeks. Before arriving, I got one of the publications that give all the questions and answers (eg Gleim) and studied that -very easy since it has all the questions that are in the exam.
Also before departing SA, I did my FAA medical with an SA doctor who is certified - much cheaper.
I used ALLATPS for the exam... paid 295USD to spend the day on their computer doing mock exams, then at the end of the day used the same computer to do it for real and got 97%. It was that simple.
I then did a B737 type rating (with another school) which also amounted to my FAA ATP checkride and got my temp ATP licence on the spot. This gave me an FAA Multi-engine ATP with B737 type rating. The flight school aranged my DOJ clearance before I arrived in the USA. Beforehand, I also summarised my flight experience in the format used by the FAA and had my logbooks examined and verified. I only did it this way because I required a US B737 type rating for the job I was applying for in the US.
So in summary, yes you can go the route of getting a PPL in exchange for your foreign licence, but my method was easier. In other words, your hours and experience are recognised and qualify you to take the flight test/checkride as long as you pass the ATP theory and have a valid medical.
The flight test was very challenging (about 2.5 hours) as was the 2.5 hour oral exam, but the flight school had me well prepared.
You can do exactly what I did, by substituting the B737 sim I used, for a seminole or seneca etc and do the FAA ATP checkride in that. I would recommend ALLATPs again, because they are established and this sort of thing is their speciality. YOU GET WHAT YOU PAY FOR. Flying in the US is very enjoyable and simple, once you become familiar with the differences. Don't go with cheaper operators to try save a few bucks, because you will end up spending more.
To get single engine land added to my licence, I simply hired a C172 for 3 hours with a CFI (have to log 3 hours) who showed me some of the required test items and then signed my logbook. I then organised a checkride in the same C172 with an examiner and got the SEL added to my ATP. Again, I was grilled about US airspace etc and then did a very stiff test in the C172 at night - all single pilot IFR. By the time we did our final landing, I was on limited panel with magnetic compass doing a circling approach of an ILS localizer backcourse.
I did my ATP and B737 type in Dallas, TX and the single engine addition in Orlando, FL. I would also recommend Phoenix, AZ.
So it can be done. Feel free to PM me for further questions.