Examples of writing systems that used characters primarily to represent sounds did come later, though similarly, it is difficult to argue that any known language is purely phonetic. As a simple example in English, take '&' — though it is not a recognised letter, neither is it punctuation, and it is unarguably a glyph in the English writing system about which one could debate whether it is ideagramatic or logographic. (But I digress.)
There are many types and varieties of phonetically representative writing systems — most of the earliest, like Linear B, are Syllabaries[2]. The earliest known surviving examples date from the mid-15C BCE, written forms of Mycaenian Greek[3].
Fully segmental writing systems (please distinguish 'fully' from 'purely', as elaborated above) are definitely younger than the rest. The earliest examples of these are often abjads and abugidas, which either eschew vowel representations or form consonant-vowel digraphs respectively[4].
The Phoenician alphabet, or more correctly the Proto-Canaanite script that was used by the Phoenicians, is the earliest surviving example of a segmental writing system — it is an abjad of which the earliest surviving examples are from roughly 1000BCE[5].
The earliest preserved writings that aren't just numbers date to about 3100BCE in Sumer[6], so predate preserved phonetic (excluding syllabic) writings by about 2000 years, and are in the order of 5000 and 3000 years old, respectively. So it's a big stretch to call it a 'recent invention', but certainly not as ancient as logographic or mnemonic writing systems.
[1]https://en.m.wikipedia.org/wiki/Toki_Pona
[2]https://en.m.wikipedia.org/wiki/Syllabary and https://en.m.wikipedia.org/wiki/Linear_B
[3]https://en.m.wikipedia.org/wiki/Mycenaean_Greek
[4]https://en.m.wikipedia.org/wiki/Abjad and https://en.m.wikipedia.org/wiki/Abugida