Curious: why do you prefer syllabaries? I think about Chinese writing systems, which additionally don't have clear word boundaries. Now you can argue that this is an independent issue (which is true) but why does this complication seemingly show up in such writing systems?
Anyway, alphabets have been profoundly successful. You bring up Turkish. It's a good example. I'm sure most here know that prior to 1929 or so Turkish used the Arabic writing system. This is also an alphabet but a more complicated one (eg vowels aren't typically written) and didn't necessarily fit the Turkish language.
So they designed a Latin writing system that is entirely phonetic it was was profoundly successful at increasing literacy rates. A completely illiterate person could be taught to read and write Turksih in a matter of months.
I compare this to Taiwan that has high school competitions to see who can find a word the fastest in a dictionary because knowledge is required of the roots and symbols. There are thousands of characters to learn in Chinese languages. As a foreigner, this will often take a decade or more. I've seen accounts of people who have spent a decade learning Chinese that still struggle to read books intended for 12 year olds.
Literacy is so transformative to one's life that I'm so on board with anything that makes that easier.
The Arabic writing system is an abjad, not an alphabet. The two kinds of system are closely related (the Phoenician script was the origin of both the Greek alphabet and the various Semitic abjads and was itself an abjad) but are not the same thing. Abjads are well-suited to Semitic languages where the vowels are less important for morphological reasons, but in Indo-European languages (like Greek) and Turkic languages (like Turkish) vowels are important in writing for comprehension. It's no surprise that switching to an alphabet aided literacy in Turkey.
Abjads are no more or less complicated than alphabets though. They're just a bad fit for Turkish.
It really depends on the language, but the main reason why syllabaries can be simpler in terms of mental load is because syllables (or moras, or something similar) is how humans naturally break down words in most languages. Consider that even in languages like English with a well-established alphabetic orthography, words are still routinely spelled out syllable by syllable.
However, phonotactics place some practical limits on that. If your language only has (C)V syllables with a fairly limited phonemic inventory - e.g. Japanese - a syllabary is small enough, and even if letter shapes for related syllables aren't consistent, is easy enough to learn. OTOH if you routinely have, say, 3-consonant clusters (as in e.g. the word SCRipt), you quickly get combinatorial explosion.
To some extent this can be mitigated by alphasyllabaries like Hangul, where syllables are clearly marked in writing but still composed of some more fundamental units. Still, you can only cram so many tiny elements into a single glyph that is still distinct. Hangul actually used to have standard syllable blocks for all combinations of three consonants in the onset, and those are already very hard to read.