Any of you players aspiring to either start with or convert to CBA without a teacher will be pulling your hair out with regard to obtaining a suitable teaching method, unless you decide to develop your own fingering patterns.
Most people these days will agree that the thumb is indispensable when it comes to right hand fingering, as it should give more possibilities across the treble keyboard. However, if we wind the clock back a couple of generations you'll see that another school of thought prevailed regarding the use of the thumb. If we consider stringed instruments like the violin it will be obvious that the player only has 4 fingers at their disposal, as the thumb is used to hold the instrument, and cannot effectively be utilised to assist the player.
With that in mind, the early diatonic "button" accordions tended to be treated in the same manner, with the thumb being used as a support for the right hand. When the chromatic button accordion was invented, the diatonic technique was generally adopted. The buttons were closely enough spaced so that an octave could be played with the first and third fingers of the right hand. Although the early CBAs were popular in other countries, method books were published in Italy and France, where these instruments became most popular. The staccato playing which consequently became popular in both countries was better emphasised with fingering patterns which excluded use of the thumb.
Then as accordion music struggled to keep up with the times, several prominent CBA players began to experiment with using all 5 right hand fingers. These included Carlo Venturi from Bologna, Italy, and also Jo Privat from Paris, France. Both players took their respective instruments and styles to new levels of proficiency, whilst traditional players like Gigi Stok from Modena, and Aimable Plouchard in France persevered with the old style. In fact in times gone by it was common for most CBA accordionists to play with 3 fingers only in both countries.
Those of us old enough to remember those players had to make a choice. I read many articles regarding the best fingering methods, but in my home country of Scotland all of the CBA teachers, who were thin on the ground even then, stipulated that the thumb was mandatory on CBA. Not surprisingly I flew in the face of all the advice I was given. I decided to go for a French CBA and play it their way. In latter times the Italians have admitted that the French are quicker without the thumb, but state that their own Carlo Venturi, who used all 5 fingers, "wasn't far behind".
These days the most common teaching methods for CBA in France advocate the use of all 5 right hand fingers, as do their Italian equivalents. In fact the most common method book in France these days is written by Richard Galliano, born in France to Italian parents. He is one of a number of modern "French" players who have shunned the quirky French CBA instruments in favour of their more solid Italian built cousins, complete with those big treble buttons.
So, we are all agreed that the thumb is indispensable then?
Check this Italian guy, Wolmer Beltrami, out. In his day thumbs were for anything you could imagine, except for playing accordion.https://www.youtube.com/watch?v=6tnkUo32zIo
Bellows straps? Reckon he'd heard of them, but they got the same importance as his right thumb!