Paul's advice sums it up. Use the textbooks as a beginner's tool, then find out what suits you best. You might even find that certain music styles are better played using different fingering. The arguments often broached with regard to CBA fingering is the cause of a lot of anxiety and lack of confidence amongst even experienced players. Modern school says get that thumb working, as there is no point in making things difficult just for the sake of it. In the days when the thumb was regarded by some CBA teachers as "the enemy", popular mainstream accordion music was probably less demanding, and four fingers were usually enough to get by on. Even some of the most complicated popular pieces like "La Migliavacca" can (rather uncomfortably) be played without the thumb, albeit by very experienced and accomplished players. However, why waste a lot of time trying that, when you have that extra finger sitting in the "sin bin" on the edge of the keyboard?
As a glaring example of the confusion that can arise, in my last post I said hit the C with finger 2 or 3, when I meant 3 or 4. The method I learned referred to the digits as
0-1-2-3-4, with 0 being the thumb, so I reverted to that terminology in one phrase then went modern in another. The number 5 would only apply in the notation I learned by if you had 6 fingers on your right hand!
Another famous pro player, Joe Rossi, lost his third finger during WW2, and had to re-learn the whole shebang with the fingers he had left. He was a top recording artiste for years. Yet another, Daniel Colin, who is of extremely small stature, devised his own fingering method so that he could make use of his thumb.
To go on, a lot of self-taught CBA players here in Scotland tend to mainly use their thumb and first two fingers across all 5 rows, and I know a pro player who plays that way, and has made several commercial recordings on cassette tape and CD. I for one wouldn't dare tell him he isn't using the correct fingering!