Yes, this is basically the argument I was making. For someone with no savings, inflation doesn't matter at all - as long as there's enough to eat.
There are several arguments I'd make as to why that isn't true. I already alluded to some earlier. But wage increases as a result of monetary supply expansion and the resulting price inflation are usually a result of several orders of effects after the fact. Therefore there has already been price inflation in the economy impacting their ability to put food on the table even when wage increases are being received.
Also, whether or not there is actual price inflation (that which CPI attempts to measure) does not mean there wasn't inflation in the economy (expanding money supply). There are many factors that come into play that determine whether or not you'll see price inflation. For example, high private debt levels, aging demographics, technology advances and productivity increases, wealth concentration, commodity oversupply, outsourcing, etc. These are all deflationary effects in the economy that can counteract an inflationary money supply. Many of these things are great things that are taking place in the economy, but whose impact on the economy is barely felt because of an expanding money supply. So when you have an inflationary money supply, that means you can have people advancing technology and productivity measures, but because of a greater supply of money (that the poor rarely are able to take advantage of), that means you have people developing these things and unable to capitalize on the greater good that these things can provide society. There should be no reason why we have such a high advancement in technology with people still working 40-50+ hour weeks just to barely put food on the table. Because we have an ever increasing supply of money, it is robbing people of today's productivity and technology advancements that could be used to provide them wealth in the future. If the supply of money were static, these deflationary forces would be felt to a much greater extent in the economy which means the work that today's lower-middle class puts in today would still have purchasing power into the future as well.
Saying that inflation doesn't matter at all ignores all the orders of effects that take place that lead up to price inflation. It is also important to distinguish between monetary inflation and price inflation of goods. One is the cause of the other. Also, even if price inflation isn't seen doesn't mean there hasn't been monetary inflation that still impacts people's lives even if it isn't directly felt in the price of goods. Saying that just because you're working 40 hours a week to earn $500 dollars and spend $500 dollars and then in the future that turns to $700 earned and spent on the same goods ignores the fact that you just spent years working 40 hours a week that went toward productivity advancements and technology improvements over time that, in all reality, should mean you shouldn't have to work as much for that same amount of money. This is ultimately where the poor lose out. Instead of them being the beneficiaries of their work, inflation robs them of this.
Meanwhile, with monetary inflation, that means that all scarce assets (stocks, real estate, precious metals, commodities, etc), rise in price and the rich who are highly invested in these asset classes can capitalize on these price increases. There is a reason why P/E ratios are out of whack and real estate prices are soaring. This is where wealth inequality is really exacerbated. The poor can't capitalize on these asset classes like the rich can. So the rich get richer, but the poor, who are really helping drive our economy with new tech and productivity, barely can get food on the table while continuing to work 40+ hours a week.
