No I am not wrong about it.
Once you have been checked once and completed your PPL you do not need a training vias or subsequent TSA clearance.
I know because I was told that by the TSA - and here's the biggie - I have returned to the US for other ratings and endorsements - declared it as my reason for visiting the US and had no hassle at all.
The training visa was for ten years - but with seven years left to run on it I was pulled out of the line and had it cancelled - this was on an ordinary (no flying) shopping trip - since that time I have travelled three times for training with out a problem.
I think you are confusing pre-TSA PPL's with those obtained afterwards (mine was in 2011)
If you think I am BS'ing you read
https://www.aopa.org/advocacy/pilots...light-training
Also you will need to read
https://www.ecfr.gov/cgi-bin/retriev...n=14y2.0.1.1.2
you are looking for what 14CFR 61.31 covers
https://www.ecfr.gov/cgi-bin/retriev...#se14.2.61_131
I don't care if there has been a long discussion on the subject - read the source info - if you think it doesn't say what I maintain, then you are wrong - I know, been there done that!