I see your confusion. The thing is that “trigonometric proof” is not a well-defined mathematical object (even in this HN thread, you can see a lot of discussion about whether the trigonometry is really essential to the proof, ways to remove it, etc). So the “establishment” idea that “There are no trigonometric proofs” (Loomis, 1927) is not a mathematical conjecture, which can be demolished by the first counterexample (the first “trigonometric proof”), but a meta-mathematical statement or social convention, where minds change slowly. Note how he says “this point of view has been
increasingly questioned in recent decades”: this is typical of the evolution of social consensus rather than of mathematical conjectures (which would quickly switch from false to true or vice-versa as soon as a proof or counterexample is found).
The first few trigonometric proofs, being very complicated, might have just had a reaction (among the very few people who even care about this question) like “yeah ok, whatever, that's just too contrived, not very interested”, but when a proof like this comes along, being more beautiful and simpler, more people will change their minds — but even now it's not guaranteed, which is why the author says “might make a few established mathematicians eat their words”.
Ultimately, all proofs are just pushing around of axioms and implications; there's no clear separation of whether a proof is “different” from another or whether it's “trigonometric”, but in this case the authors say their proof “is based on a fundamental result in trigonometry—the Law of Sines” and it seems pretty easy to believe that that is how they came up with the proof (so it seems fair to call it a trigonometric proof even if that can be got rid of).
[PS: I just found that some scans of Loomis's book are online: https://personal.math.ubc.ca/~cass/Euclid/java/html/L.pdf (1927), https://files.eric.ed.gov/fulltext/ED037335.pdf / https://www.lapasserelle.com/documents/Pythagorean_Propositi... (1940 second edition) — see the foreword where he says “Fifth, that no trigonometric proof is possible”, elaborated on p. 193(1e)/244(2e) in section called “No trigonometric proofs”.]