The Greek alphabet was derived from the Phoenician alphabet, and the fact that it replaced the local writing and spread quickly suggests that a simpler technique of writing can win. I think the trend you see is the most common behavior.
But the evolution of written languages is not a simple process. It is often influenced by political choice. E.g. Hangul for Korean, simplification laws in Germany or France.
I can think of few exceptions where the simplification of a written language was rejected. The Sumerian ideograms were simplified after their introduction circa -3300. The writing system was adapted to the Akkadian language which was the dominant spoken languages for centuries. It mixed pure ideograms with double readings and syllabic symbols (a bit like Japanese). But after 900 BC, when neo-Assyrian ruled, Akkadian was declining, as well as writing on clay. There was a clear tendency to complexify the tablets: for example adding more symbols or reviving archaic expressions. Akkadian was the language of the elites, and simplifying it would "degrade" it.
Disclaimer: I have been interested in ancient writings, with a particular attention to Mesopotamia, but I'm not an expert.