What a frustrating article to read.
tldr;
"Why Are Alphabet Letters Arranged How They Are?"
Because the Greeks modified the Phoenician alphabet a bit, and used that order.
"Well...why was the Phoenician alphabet arranged the way it was?"
<no answer>
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The Latin alphabet lost a few Greek letters, but retained "F", which existed in older Greek alphabets but was lost in newer alphabets. Latin changed the pronunciation of a few letters and later added letters, at the end of the alphabet, except for "G" (which replaced an abandoned Greek letter).
Before that, the Greeks took the Phoenician alphabet, which had 22 letters ending in "R", "S" and "T".
The Greeks lost a few Phoenician letters and added new letters after "T".
The Phoenicians inherited the alphabet from other closely related Semitic people (very little is known about those), but the Phoenician language was simplified during time and the Phoenician alphabet lost 5 letters from the older alphabet, which had 27 letters.
I am not aware of an older alphabet whose order is known, besides the old 27-letter ancestor of the Phoenician alphabet.
Most evidence about the letter order of the old 27-letter alphabet comes from the Ugaritic alphabet, which was also derived from it, like the Phoenician, but which replaced the graphic signs used by the older alphabet with cuneiform signs, nevertheless retaining the names and the order of the original 27 letters. The Ugaritic alphabet added 3 new letters after the original 27 letters.
The names of the first 27 letters of the Ugaritic alphabet can be seen in the Unicode database (from 0x10380 to 0x1039A).
Many school exercises with the Ugaritic alphabet have been unearthed, so there is no doubt about the letter order.
I have forgotten to mention that my point was that all known alphabets derive their order from an ancient 27-letter Semitic alphabet, so any hypotheses about the origin of the letter order must discuss the order of the original 27 letters, not the order of the letters of more recent alphabets, which just copied the original order, then deleting some letters and adding new letters, usually at the end of the alphabet.
(My guess) IMHO, letters are originated from pictographs, as way to script pronunciation of an unknown word (e.g. name of a foreigner) by first sound of word represented by a pictograph. Later, they are adopted to represent foreign speech as is (e.g. to learn it), then letters are adopted by foreigners for their own use.
To remember all these symbols and their pronunciation, a mnemonic is necessary: a well known short speech, and list of letters for first sound of each word of this speech. From there, alphabet is derived naturally.