My guess (as an English-only-player): Anglo is the most intuitive system, Duet the most neatly organized - and the English? Well, it has a very special "logic" with it ... , thus facilitating some extra "fluidity" which might be harder to acquire with other systems...
This is nicely put, but I wonder if the intuitiveness, the organisation and the logic are not perhaps chatacteristics of the player, or maybe of the collective of instrument and player.
Sure, the Anglo as such is intuitive to use, if it matches your intuition (it matches mine!). And the duets are neatly arranged: the Maccann for accessing the notes quickly, the Crane for finding them methodically. The logic of the EC is much vaunted; to me it's a "logical" extension of staff notation conventions (as you explain).
But the fact is that, whatever system you learn (and I mean learn, not mess about with), eventually things come to you intuitively; you invent an "organisation" behind the system, even if there isn't one; and you'll work out a "logical" basis for improvising in places you haven't been to before.
When you've got all this, and mastered the method of making chords and chord progressions, you're ready for song accompaniment. Which instrument you choose - a specific concertina system or a stringed or keyboard instrument - is not so important. What you can learn about song accompaniment can be applied to most instruments - most definitely to any instrument that you can play well!