.. the problem I'm having now is to see how his argument can ever have been believed, by him or by anyone elseOr was Aristotle's notion something quite different, such that his view that 'it' is not composed of indivisibles is perfectly consistent. In which case, what on earth was his notion?
I think I see a way out (noting carefully that I am not a mathematician, and this is just my two cents). Connolley starts with the idea that the continuum is just the real numbers between two points (say 0 and 1). If that's what the continuum is, i.e. if it is just those numbers, then it's surreal to ask whether it is composed solely of indivisibles, i.e. composed of numbers. If that's how you define it, it's an absurd question. And even more absurd to argue that is isn't composed of numbers at all. That would be like concluding that bachelors are married men.
But we don't have to start with that idea at all. Suppose we characterise what-is-continuous as that which is divisible into parts, and which after any finite number of such divisions leaves parts which are continuous themselves. Then it is an open question whether such divisibility could be completed or not. Clearly there would have to be an infinite number of such divisions, since by definition any finite division leaves continuous parts which can be further subdivided. And if that were possible, i.e. if it were possible to complete the process, then by definition of 'complete', what was left over would be indivisible.
So perhaps it is coherent to hold that Aristotle agrees with the moderns in a defining characteristic of the continuum (i.e. infinite divisibility into parts), but disagrees over the accidental property of whether the process of divisibility can be completed. And disagrees, of course, that it is an accidental property at all, for he holds that the impossibility of completion can be proved by logical means, and is thus an essential property.