this argument gets made a lot (for instance, re: the lower capital gains rate) because proponents want it to be true and its flaws are not immediately obvious, but it's unambiguously mistaken.
first, esteem from wealth is relative, and as long as there is a spectrum of wealth, the wealthiest will derive maximal esteem regardless of the absolute value of wealth. and that's the primary driver for the ambitious and greedy, not putting resources to highest and best use (which is at best a side effect). greater equality doesn't change that equation one iota.
second, it's plain to see that large accumulations of wealth are fairly unproductive, simply illustrated by the localized real estate bubbles driven by both wealth seeking safe harbor and PE-backed landlords seeking unproductive rents.
in fact, there seems to be a central optimum. relative to now, wider dispersion of wealth (less wealth disparity) seems to provide the best bang-for-buck when it comes to efficiency, putting more potential to work rather than mostly relegating it to the menial sidelines.
They are absolutely bound together. Ensuring that every person has the same power in the political process means that everyone has the time and energy to pay attention and be educated on what's going on the world, and that nobody has so much wealth that they can use it to buy influence. The trend towards more inequality while the working class has less and less political power is painfully obvious, especially in the United States.
That cost is insignificant when the ruling nomenclature is a relatively small portion of the population and when basic workers rights are not respected. Especially in manufacturing stuff on a pipeline in huge quantities, a totalitarian system can be much more effective output-wise, at the cost of disregarding any environmental and human condition externalities.
This cost can be extremely high, as the autocrat has to pay off enough people to ensure that rivals cannot achieve an advantage.
Briefly, rising incomes under a non-democratic regime lead to a local maximum of growth, as totalitarianism is inextricably linked with economically inefficient institutions. As an autocrat, your top priority is to compensate those who keep you in power, which requires wealth extraction from the productive; this disincentivises efficient production. Unfortunately, nothing about rising wealth will necessarily lead to regime liberalization.
"Reducing inequality" is very different from democracy, and would probably lead to reduced economic efficiency, as it necessarily reduces the incentives for efficient production.