Did Mao want to convert written Chinese into a phonetic system?

Yes.

20th of December, 1977 edition of People's Daily, on the nation-wide announcement of the Second Round of Simplified Chinese Characters. The prominent quotation on the top right is as follows:

毛主席語錄

文字必須改革，要走世界文字共同的拼音方向。

Quotations from Chairman Mao

The writing system must be reformed; we must move in the direction of a globally unified phonetic spelling system.

The movement towards a phonetic system, like the further "Simplification" of Chinese, failed to gain support among a literate population, and is now on permanent hiatus.

I fear you're missing the point when you claim that "phonetic writing of Chinese is possible". Sure, spelling Italian words with Spanish cognates is possible, but at that point the word possible becomes something of very little utility.

Updated with details from comments:

"Chinese" is a language family, and most phonetic scripts developed for Chinese are a realisation of only one of the languages or a group of closely-related languages in the family. For example,

• Jyutping is only for Cantonese;
• Pe̍h-ōe-jī is only for the Minnan branch of Sinitic.

Likewise, the nationwide Romanisation scheme in China is called Pinyin, and is only for a national standard of Mandarin Chinese (unhelpfully for this topic, the national standard is called "Standard Chinese"), not any other varieties of Chinese. To claim that it is possible for phonetic script to represent "Chinese" is disingenuous for multiple reasons:

• There's an implicit assertion that "Chinese" = "Mandarin", when Chinese is a large group of languages including Mandarin;
• Chinese people don't strictly read and write "Mandarin"; the actual literature that people enjoy comes most often as a dialect continuum, with a smooth transition between the national standard, local topolects, and Classical Chinese;

• The examples given in the question to demonstrate that Mandarin speakers can't read non-Mandarin topolects, even if they are written in characters,

  “佢叻D。”

  “你袂使佮伊出去𨑨迌。”

  are red herrings - they are designed to distract people from the fact that substantial bodies of literature, like most Wikipedia articles written in non-Mandarin languages (Gan Wikipedia, Yue Wikipedia, Wu Wikipedia) are majorly accessible for Chinese-literate people who don't actually speak those languages, while Minnan Wikipedia, written in a phonetic script, is completely inaccessible for non-Minnan speakers.

It seems to me that the only way for a statement like "it is possible for phonetic script to represent Chinese" to make sense is to restrict "Chinese" to mean "Mandarin/Cantonese/Wu" (completely disallowing non-Mandarin/Cantonese/Wu expressions, respectively) and to impede usage of Classical Chinese vocabulary. If this is the case, I'm afraid that this is a complete misunderstanding of how Chinese-literate people use Chinese characters.

The critical issue was that, for the Chinese Communist Party, the priority was the development of a Common Spoken Language nationwide, which turned out to be Mandarin based on the Peking/Beijing dialect. This then needed to be taught to those who used other dialects or other mutually unintelligible Chinese spoken languages. Classical written Chinese was unsuitable for teaching speech either to those who were literate in it or to the larger number who were not.

In 1951, Mao issued a directive on reform of the Chinese Writing System saying it should develop in the direction of a European-type alphabet. Attempts by the Chinese Writing System Reform Committee to develop a local alphabet failed and by 1957 they had settled on and promoted the Latin-based pinyin. This was endorsed by the People's General Assembly the next year, so textbooks and newspapers started to appear. Pinyin turned out to be effective in teaching speech both to Chinese people and to foreigners, and for communications such as telegraphy, Morse code and Braille. It met some resistance as the main writing system from those already literate in written Chinese, and other efforts were put into simplifying written Chinese characters.

The Cultural Revolution (which Mao launched) from 1966 to 1976 effectively blocked any further progress. By the time it was over, Mao was dead.

So alphabetical pinyin has become an auxiliary representation rather than the primary written form of Chinese, largely because Mao had other political priorities.