>The poor tend not to have much savings, by definition, which means that all this win goes to the already wealthy.

This is not true - saving rates were higher in the past before irresponsible monetary policy took over after the erosion of the gold standard.

>Your argument is simply that your dollars should be worth more later because you got your dollar first.

This is exactly what happens in an inflationary monetary regime, althought the difference is that the distribution of purchasing power increases is not homogenous in an economy rather it is concentrated on those who are either politically or economically well-connected, usually asset holders.

>Your argument is simply that your dollars should be worth more later because you got your dollar first.

How about not controlling the value of money and instead letting it reflect the general purchasing power in an economy?