I thought I would open this post up to experienced C system players, and just ask how you approach this song. Regarding bar 3, which is after the intro. When I first picked up the instrument, I went straight to row 4, and hit the C, D trill with fingers 2 and 3, as it seemed the clearest approach. Since reading a few texts it seems I should be hitting the C, D trill with fingers 4 and 5. How would you approach this song?
I cannot claim to be an "experienced" C-system player, but I would do the same as you did. The 2 and 3 fingers are naturally stronger, and unless you've spent significant training on your other fingers' muscles (say, via Hanon exercises), it's just much easier to use them.
I've only been playing since last March, and like you I've had no teacher, but I am firmly convinced there's no "right" approach to this sort of thing. Initially I had decided I would try to limit all of my playing to the outermost 3 rows, as OuijaBoard does for melodies. My reasoning was that if I limit myself to the outermost 3 in that way, then I can easily transpose to any other key without any practice whatsoever, simply by shifting where my fingers are at. And also, if I found myself playing a 3-row-only instrument, it wouldn't be a big adjustment. However I quickly decided that this was a poor choice for me. Among other things, as OuijaBoard hints, it really doesn't work well for chording at all - some fingerings are just way too awkward and impractical if you're trying to play chords and limiting things to just the outer 3. For that same reason, I have no interest in preparing myself to be "instantly acclimatized" for a 3-row instrument
For myself, I'm roughly equally interested in improvising and playing by ear, as I am to play music (I'm an accomplished pianist, and so learning to read sheet music is not something I need further training in). In order to improvise, I find that it works well for me to play RH melodies in such a way that if I wanted to add chording under the current melody, it would fall fairly readily under my fingers, even though I may actually only be playing a melody. For me this mainly means playing in roughly C-major scale fingering, and shifting that fingering to whatever key I'm actually playing - because I find that, while D major (beginning on row 3) is probably the most fluid-feeling for melodies (fewest skips), the 3-row scale patterns that start with the outermost row lend themselves the best to forming natural positions for the most commonly-needed chords.
I don't limit myself to playing within that scale pattern, but it forms the foundation, the default I go to for everything I play (except that, when sightreading anything with much in the way of accidentals or more than a couple sharps or flats, I stick to the outermost 3 rows, as it's easier for me to think of where specific notes are in that range where I practice - in other situations I'm thinking more of intervals than specific notes, which lets me range). Because it allows me to most easily add chording to something, later. So, since I'm already ranging the 5 rows due to this "baseline" practice, I am a very firm believer in using whichever rows offer the most fluid passages, at all times. Another reason I hold this belief so firmly (for my own practice and performance), is that, given all the many things that can threaten your performance, you really shouldn't make any aspect of it harder than it needs to be. Do what encourages fluidity, above pretty much all other considerations.
Aside from resolving awkwardness in fingerings, I have also noticed that sticking too strictly to a given 3-row range can also force my wrist to turn for the position, which I view as a thing to be avoided if practical, especially in brisk passages, as IMO that wasted motion can make breaks or mistakes more likely.
I don't believe these policies are sacrilegious, though. My impression is that. on minute questions of technique of exactly this sort, universal agreement does not exist. This is why even among the 3 major scale patterns, you can find something like half a dozen or more different recommendations for fingerings, for
each of those patterns. Most of which usually seem outlandish to me, but apparently my preferred fingerings (Anghazi's, I believe) provoke that same reaction in others. You may hear people insist that the thumb be avoided wherever possible in fingering a CBA, and many others (myself included) will pronounce that to be BS. (I strongly suspect that advice makes much more sense on a B-system, where the chromatic scale is naturally played 2-3-4, 2-3-4). There simply does not appear to be a "sacrament" with CBAs to violate: the schools of thought are too varied.
But, even if it were "heretical", I wouldn't care overmuch, unless someone could convince me of
why, specifically. It's not enough for me to know "it isn't done this way", I must know what the consequences are for disobedience. Because, often, the values of those who framed such traditions and policies long ago, may not be the same values I myself hold today. And, honestly, I believe the traditional accordion, especially of the Western European tradition, is itself, in core aspects of its very intent and design, "heretical" in several respects. Particularly the Stradella system: "You can play a note, but you can't indicate a particular octave." "You can play a triad, but you can't choose its form." And, for the treble (mostly in W.E. traditions), "Oh, you want a tremolo effect? We'll simulate that by playing multiples of the same note, out of tune to each other". Every one of these ideas is anathema to certain classical schools of musical thought. The traditional design of an accordion really doesn't lend itself to elitism or purism, IMO (though the advent of free bass closes this gap somewhat).
So IMO it's not worth it to worry about adhering to particular notions of "should do", until you understand why in fact you should do them. Because (a) they often are disputed (probably do pay some caution to any that aren't), and (b) even if they weren't, you should do what you believe makes things run and sound the best.
Anyway, forgive my ramblings, and I hope you find it useful!