First of all, there are some assumptions in your question that don't really square too well with the reality. It's kinda a loaded question, so let me first try to clear up some of that.

For starters, Mao advocated for moving to a phonetic script in 1951. This is the same time that a standardisation of simplified characters was called for, but that's not the same thing as the CCP throwing out hundreds of the most complex characters. The simplifications weren't a CCP invention, and the vast majority of those which were made official in the 1964 Scheme of Simplified Chinese Characters were already widely used, some even coming from simplifications in non-Mandarin languages.

Reforming the writing system, as well as the standard speech, was something that has been pushed for well before the founding of the PRC. For example in 1935, in pre-communist Republican China, the Ministry of Education introduced a simplification scheme of 324 characters. Other private efforts were also ongoing throughout the decade. And it's also not the case that they simplified the most common characters. The choices had more to do with characters that were in frequent use and in many cases ones that already had widely used vernacular simplifications, many of which are also in use in Taiwan which never saw any of the PRC reforms and where traditional characters are still the norm.

But back to your main question: Why didn't they make the switch? Let's set aside the notion that it's "the next logical step", because it's not (edit: see reply to /u/MooseFlyer about this point). But beyond that, the reality was just that it would be a huge amount of effort to do, and at the time when people were most vocal about it, other things were coming up that prevented further exploration. For example in the 1930s many of the language planning efforts of the Republican government were put on hold due to the impending war with Japan. In the 1960s when some of the changes first started in the 1950s were coming into effect, there were once again other things that needed attention and took precedence over writing reform.

You may be interested to know that in the 1970s, characters were further simplified, though it was poorly received and was revoked shortly after. Already Chinese is a fairly phonetic script, and the second simplification scheme pushed it further in that direction. It just didn't take.

So today, the real reason why there's not a switch is because it's neither necessary nor desired by those most affected.

• Baldauf, Richard B., and Robert B. Kaplan. Language Planning and Policy in Asia. Bristol, UK: Multilingual Matters, 2008.

• Zhao, Shouhui, and Richard B. Baldauf. Planning Chinese Characters: Reaction, Evolution or Revolution? New York: Springer, 2007.

• Zhou, Minglang, and Hongkai Sun. Language Policy in the People's Republic of China: Theory and Practice since 1949. Boston: Kluwer Academic, 2004.