I would say go for Cortot's method if you want to play like Cortot.
Go for Czerny if you want to play like Czerny played.
Czerny could, as a teenager, already play all of his friend and teacher Beethoven's piano works by memory. Beethoven preferred him of course to play with the score, and they both lived in the days before recordings existed. But Czerny absolutely hated playing wrong notes. Cortot didn't seem to mind. I'd go for Czerny myself. He was, after all, Liszt's teacher. If Liszt could learn from Czerny, I think we all could.
Surely Czerny works. It is a proven method and he did indeed train many fine pianists, including several ones who became important teachers. I think with Czerny, if a teacher provides the insight in the 'how it;s done' to supplement the practical side, it works great.
It just depends on your preferred way of aquiring technique in respect to that. Also it matters IMO if you're an autodidactic or a pupil. I assume he is the former, because otherwise I thought he would have had a teacher who could answer this question.
Because Cortot offers many valuable written material too, I found his work to be easier to use as a reference guide at home and while learning all by myself. I am an autodidactic except for a few years in my youth, you see, and when I did go to music school earlier my teacher gave me Czerny etudes as homework but he never bothered to tell anything about HOW it works. He should have, but back then I didn't realise what was keeping me from understanding what I was doing.
In regard to your comment on Cortot's wrong notes;
in Czerny's time technical perfection was easier with lighter keyboard actions, and the tempi used were way slower. Beethoven organised a concert at which he premiered the 5th symphony, the show lasted 4 hours but typical recordings of the same programme now take little over 2 hours. Coincidence? Or Liszt played the Hammerklavier according to his diary in about an hour, instead of 40-50 minutes. Coincidence?
Cortot was not just a superb pianist, he was always playing frightening tempi in fast passages, being on the cutting edge even in his later years, and seemed to enjoy taking risks, which makes his records sound unlike anything I ever heard, like he was improvising everything. Surely his playing contains more wrong notes, not helped by his outright refusal to have recordings edited together from multiple takes.
But what a technique he did have! Vladimir Horowitz was obsessed with finding out how he did the things he did, like the legendary 1919 acoustic recording of Saint-Saëns' Etude en Forme d'une Valse.
I'm sure Cortot would have been able to play technically perfectly had he taken less risks. But it would have also meant less spontaneous improvisation-like playing.
Hope this makes it a bit more clear why I prefer Cortot. Perhaps I should have left out the word 'instead'... I don't want to dissuade anyone from using the Czerny method, but I think it's worth trying Cortot because I think it offers a few significant advantages, especially when self-taught.
