Your mistake is in assuming that a dollar will provide the same demand if given to a wealthy person as it would if given to a poor person.. The poor person's propensity to spend their dollar is much greater than that of the wealthy person.

In a capitalist system money moves up the chain and only its velocity is changed by distribution. Spending on the poor increases velocity immensely in comparison to allowing the wealthy to keep what naturally flows to them anyway.

Conversely, allowing the wealthy to keep their money has no effect on the ability of the currency issuing government to spend on the poor and is not inflationary as long as proper oversight of investment and political contributions is in place to avoid/punish detrimental spending of wealth and rent seeking. Jeff Bezos isn't going to cause the price of bread to increase as long as he can't corner the wheat market or purchase a monopoly position in bakeries.