All of this is true, but I can learn that Spotify romanizes Τα παιδιά του Πειραιά as Ta paidiá tou Piraiá (or whatever) and search for that, whereas I can't type the actual name for love nor money.
I'm not saying this is necessarily a good design decision, just pointing out what might motivate this design
While I don't doubt your knowledge, note that many aspects of European language, including English, and including the European alphabets, are in a sense transliterations of Greek.
Certainly, and there isn't a canonical transliteration there either. There is some consistency, but the primary goal was that the words look good in the target language, which isn't a concern for me when I want to type my language in a keyboard that doesn't have Greek letters. For that case, multiple transliterations per letter exist, including some that use numbers (e.g. θέλω = thelo, thelw, 8elv, and any combination of those).
Ah, yeah, I agree that the way that Greek words got transliterated is actually very interesting (doubly so when you speak Greek and can see the intent behind it).