The comments in this post are in response to a particular individual's request for assistance. The advice I would give to an individual with different experience and/or desires would be different. Will be different, if other individuals ask for similar advice, which I hope they will do, and under this Topic.
No it doesn't, unless you want to be indistinguishable from all the others who play those instruments. Robert Harbron has done some beautiful arrangements of melodeon and French accordion pieces on English concertina. I've heard excellent tangos on anglo, and Scott Joplin pieces on both anglo and English which sounded as if they were originally written for those instruments. The possibilities with any concertina will be limited more by your imagination than by the instrument itself.
I am a serious quasi-professional mandolin player. But I've been curious about concertina. I love the sonic texture it gives to music. I am, however, confused about what instrument I should even be trying to test/rent prior to taking the $$$ plunge. The music I play includes a pastiche of bluegrass, contradance/old-time, country, Americana (a cover of The Iguanas aches for the sound of something with a bellows), Celtic and gypsy jazz. Perhaps I have been listening to too much accordion lately, but I would also like to play what I can only call global cafe music -- all of which has some sort of squeeze instrument in the mix. This includes French musettes, Italian cafe music, Argentinian and Brazilian music, lots of Piazzolla. And I would also love to learn some Nortenos. This obviously means I need accordion and bandoneon.
Not necessarily. Your own best bet may be the English. A standard treble English is in many ways similar to a mandolin: same lowest note and a 3½-octave range; fully chromatic; generally good for chords or melody, or melody with simple harmonies, but not for vamping chords against melody; etc. In fact, much of what you do on the mandolin might transfer comfortably note-for-note onto the English concertina, where you could then modify it to suit your own tastes and the idiosyncracies of the concertina. Such a "jump start" on your learning process could be worth a great deal.
It seems, based on what I have read thus far, that I might be best off thinking about a duet-style instrument.
The duets are a different kettle of fish entirely, with the two hands potentially independent, as in the piano. If that's the direction you want to go, that's cool, but keep in mind that if you do that you will not only be adding the concertina sound, you will be forcing yourself to learn an entirely different genre of arrangement, accompaniment, and thinking about music. If you do want to head that way, you might even be better off learning to play piano or piano accordion first, since there are much more extensive materials for learning different styles on those instruments. Then once you're comfortable with the separated-hands concept you could try applying it to a concertina.
By the way, there's nothing wrong (except possibly expense) with learning to play more than one kind of concertina. The English, anglo, and duet are as different as the mandolin, banjo, and guitar, and many folks play two or more different stringed instruments.
Standard notation won't be a problem. The problem will be finding worthwhile teaching materials, no matter what type of concertina you take up. In general, the tutors that use standard notation also teach what they use. But none really go beyond the basics of either notation or playing. (Allan Atlas' recent book is an exception, but I understand it starts assuming that you're already competent at both.)
What from there? I assume apart from workshops, I will have to teach myself. Will this be difficult without significant piano training and calcified standard notation reading skills (tablature ... where was it when I began playing guitar!)?
And it's mostly for the anglo. Allan Atlas' book is an exception on both counts, but I believe it also concentrates on a particular genre. Those few players doing the other kinds of music you're interested in don't seem to be publishing tutorial material or even arrangements. You'll have to be creative.
Most of the information out there seems to deal with playing in a Celtic or Morris setting only.