Pat,
I'm going back many years since I bought the Anzaghi method. I had been trying to play C system CBA for a while just by picking the box up and listening to records. Fingering was whatever I could manage as I hadn't seen anybody else play one.
I soon realised that I would need to find a book somewhere, and bought the Anzaghi method as it was the only one in the shop. From memory it did what most CBA method books did and encouraged a single fingering across all 5 rows covering the major and minor keys in all keys. That's absolutely fine for most of the styles around, so what was my problem?
The style I played was French musette, with a 4 row instrument, where the playing style at the time I was learning required prolific use of staccato, plus the requirement to learn 3 different fingerings so that most of the notes could be played on the outside 3 rows. French musette was essentially a folk style (it isn't now) comprised of simple tunes mainly for the entertainment of dancers. Also, the way the older musette pieces were written, most of them could be played using three fingers, with the thumb being used as a sort of stabiliser whilst you did all the fancy button rattles and arpeggios with your 1st, 2nd, and 3rd fingers, using the pinky as and when required. That's just the way the French method books were, but from the late 1970s onward they have succumbed to the Italian "thumbs on" tuition methods as per Michel Lorin, Manu Maugain, and most notably Richard Galliano.
The new style of French musette concentrates on fast legato runs, coupled with jazzy progressions and embellishments, so the more modern learning methods are perhaps more appropriate.
I personally prefer the older style and that's why I decided to abandon the Anzaghi method. It wasn't right for me, as I found that the CBA fingering made too much use of the thumb.
Many Italian CBA players have probably never seen any other books other than Anzaghi's and they can play just about anything you throw at them.
An old Scottish CBA teacher once told me that learning the Ferrero method, which I did, was like going into a boxing match with one hand tied behind your back. I certainly had to learn how to adapt to not using the thumb, but I believe it made me a better player by forcing me to use the pinky and stay on the outside 3 rows. However, following a bad hand injury I now have had to adapt my fingering slightly and now make more use of my thumb.
Basically if the Anzaghi method is suiting you then keep with it as if you change to another you'll lose months if not years re-learning the whole shebang.