To start with just one of your assertions, I don't think "real" direct democracy is likely to reduce inequality. As one example, California Prop 13 was passed by direct ballot and many would point to it as a major _cause_ of inequality.

People vote mostly of out self interest, but mediating their expression of self interest through elected representatives enables pragmatic compromises that make the most good for the most people. Ultimately, I view representative democracy as a recognition that the political process will always depend on a small number of powerful people, so it's better if those people have some accountability to the people. Direct democracy circumvents that political process, forcing representatives' hands and blocking compromise. The result is stuff like Brexit.

I do think there is a place for direct democracy, but it is very limited. Historically it has been useful in affecting constitutional change e.g. the 1967 Australian referendum to grant citizenship to Aboriginals.

There are also ideological factors at play here. When GP is arguing for direct democracy they're not arguing for it in isolation, they're arguing for it as opposed to the political-economic system of today which they see as under the thumb of capital, which in part is due to the representative model. But simply removing the representative model won't rid us of capital's totality, nor its mechanisms (inequality through equality etc.)