Dixon (the Australianist) has claimed that the Phoenician/Canaanite script is the ultimate source of all known alphabetic (purely essentially-phonemic) scripts on Earth; all other scripts are not alphabetic. But he claimed this notion as fact more than a quarter of a century ago. Is this statement still well-established as true?
This may or may not be true, depending on what is meant by "ultimate source": are we talking about specific letter shapes, or just the abstract principle of an alphabet? If the former, no; if the latter, probably yes.
Most alphabets in existence (I'm using the term in its broadest sense to include abjads and abugidas) do straightforwardly descend from the Phoenician script, though often with modifications not only of the letter forms but of the system itself: e.g. the Greek innovation of consistently writing out vowels. One case where there is some controversy is the Brahmi script of India, from which modern south and southeast Asian scripts are descended, but even there the majority opinion is that it derives from some Semitic alphabet (whether or not this was specifically Phoenician).
However, there are also alphabets which were deliberate inventions rather than modifications of an existing model: Hangul, for example. The grapheme shapes of Hangul have nothing to do with those of any Phoenician-derived script, so in that sense it isn't a descendant of Phoenician. However, even in these cases, the idea of using an alphabet (rather than, say, a syllabary or a logographic script) seems to have been inspired by some existing alphabet to which the inventors were exposed, which would ultimately have been Phoenician-descended. So in that more abstract sense, all alphabets, as far as I know, are ultimately traceable to Phoenician (or else to another related Semitic alphabet, since Phoenician was not the first of these).
Old Persian cuneiform script is an alphabet, not a syllabary. Despite a superficial resemblance to Babylonian cuneiform, all attempts to derive it from a Babylonian, or any other, model, have failed. It seems to have been invented ad hoc.
Good discussion here: P. Lecoq, 'Le problème de l'écriture cunéiforme vieux-perse', Acta Iranica 1ère série, 3, 1974, 25-107.
Second objection: The South Arabian alphabet (in turn the parent of the Ethiopian alphabets) does not derive from "Phoenician/Canaanite script". Rather the North-West Semitic ("Phoenician/Canaanite") and South Semitic scripts have a common, as yet unknown, ancestor. This can be seen from the extra consonants shared by South Arabian and Ugaritic scripts, but missing in Phoenician and Canaanite scripts.