Inflation has an entirely different meaning, and causation, than it did prior to ’71 when we left the archaic gold standard and fixed exchange rates inflicted upon us post WWII by the Marshall plan. Prior to that the currency had a direct relationship to a mostly fixed commodity supply so that any increase in the currency supply had immediate impact on the value of all existing currency. That is no longer the case, even though most economists and our government refuse to acknowledge the fact.
The primary driver of inflation is no longer the money supply, but the availability of goods and services that the currency represents (using money and currency interchangeably here). It is only when the potential of the market to offer goods and services is diminished that inflating the currency supply inflates CPI. This is a simple bidding up of prices from a larger pool of currency to purchase limited goods and services. We are so far from such a condition currently that inflation from that perspective can be ignored in macroeconomics. Microeconomics in individual products may have more impact on inflation. IE: meats, as ethanol production competes with livestock for feed grain, making production costs skyrocket. These, however, are market variables that will work themselves out via innovation and market demand adjustments over time and not relevant to a discussion of currency supply.
You are correct that economic gains have primarily gone into the financial sector where they don’t impact the Main St economy except to sequester the currency. This could be easily overcome by a) targeted taxation that either collects taxes from wealth or modifies its behavior in the market, or b) injection of currency into the Main St economy at as low a level as possible to provide sufficient velocity. A hike in the minimum wage would be a help, but would also directly impact CPI very quickly. A better approach would be a New Deal type infrastructure initiative accomplished without the usual selling of debt instruments for funding. This could even be tied to a guaranteed job program that makes government the employer of last resort and takes on social needs with labor exchanged for minimum wage. This would also require employers to step up their game in both wages and working conditions, making the minimum wage irrelevant in the private sector except as a baseline.
To accomplish any of this we must first bring our government in line with the reality of a sovereign currency and the irrelevance of “debt” when it is denominated in that same currency. The federal government, as the sole issuer of the currency, needs neither taxation nor debt to create currency. When we look at a spreadsheet of the “economy”, and not just the budget, we see the obvious connection between deficits/debt and private sector reserves. Once that is realized we can see the insanity of injecting a dollar into the economy, taxing back ninety cents, and pointing to the dime (deficit) as the source of all of our problems. Where, and how, we inject currency, not the quantity, will determine our success or failure, as well as inflation.