My view concurs with Englishal above.
Lot of people have looked into this. Now, they could ALL be wrong, I don't know.
Loads of Brits etc have trained at small Part 61 schools over many years, before things got tightened post 9/11 and you needed TSA+Visa. Now, to get the M1 you need an I-20 which can come only from a school in SEVIS and only Part 141 schools can be members...... but that should be the only reason Part 141 comes into it.
Moreover, it is entirely possible for a foreigner to sort out a training request on the TSA website for a Part 61 school (I did it myself) and only later discover that they cannot deliver the I-20. You lose the TSA fee, and a lot of time. I would have thought that if foreigners could not train at Part 61 schools, the TSA website would disallow this.
Another thing is training outside the USA. Theoretically, this is subject to TSA also, and indeed various instructors in Europe are on the TSA website pulldown menu. These are not Part 61/141 - the concept is meaningless outside the USA.
In years past, you popped on a plane, went to Florida on a holiday, and walked into a school, and got your training done. No questions asked. The 18hrs/wk threshold has "always" been there so presumably everybody was relying on that to avoid the Visa, but one would rarely fly more than 18hrs/week anyway - it's hard enough work to do 2hrs/day 6 days/week which is only 12hrs/week. Even with ground school (another grey area - does this count?) you would not reach 18hrs/week. So if anybody checked, you would be in the clear without a Visa, and I believe this is still the case today.
What we have however is U.S. Immigration employing dumb underpaid untrained officials who can barely read and write and nobody wants to be the one who signs the papers for the next hijacker, so they write their own rules. And most of the school don't understand the rules, don't want to understand the rules, TBH many of them can barely string an email together, so they just say you need an M1 and that's it. The staff at the US Embassy isn't much more literate and the more times you ask them the more different answers you get.
That's how much FAA training works. With everybody inventing their own rules, it's never been an especially transparent process. I could point you to one hilarious website in Europe....