that training can commence as soon as the TSA have acknowledged receipt of the relevant information.
Whether this is true depends on the type of training you're going to do. Details on the TSA website, but for some types of training you can start as soon as the TSA acknowledges receipt of your fingerprints, and other types you can only start once the TSA has finished the background check.
The TSA simply state that if the individual is found to not have the correct visa they will then take action.
That's not quite true, I think. This is from the FAQ on the TSA website:
What happens if I take flight training without having the proper visa?
AFSP may cancel an applicant's flight training request if AFSP becomes aware the candidate is intending to take flight training without the appropriate immigration status. If you have questions about a cancelled request, you may call AFSP at (571) 227-4544.
So what you need is an "appropriate immigration status". I still believe that being admitted under the terms of the VWP would be an "appropriate immigration status" as long as you abide by the rules and regulations of the VWP. And one of those rules is that your primary reason is short-term business or tourism, not flight training.
Also, if you look at the Interim Final Rule (
http://www.tsa.gov/assets/pdf/IFR_Alien_Pilot.pdf) and search for the word visa, you will notice that it never mentions the word visa as a hard requirement. The word is used, but in the context of submitting your passport and visa details if applicable, and so forth. So TSA simply defers to the USCIS rules on visa requirements.
Of course, this is all very easy to type from the comfort of my own home. It's a different matter trying to convince a TSA auditor that you are following USCIS rules to the letter when you are found training under the VWP. So despite all this: play it safe and if there's any doubt, get a visa.