Historically, every attempt to eliminate inequality has ended VERY badly. Look at the amount of death and suffering communism caused in the 20th century. The Left is too quick to discount this as "well, it just wasn't done right." IMO, the fundamental flaw is the concentration of power required to eliminate economic inequality creates a new worser form of political inequality (the politburo vs. the people).
To be clear, I'm not arguing for an absolutist position on economic systems -- I'm not an ideologue. A sprinkling of socialist programs can offset the worst excesses of capitalism. Those that work hard and smart, and get lucky, should be rewarded with lifestyle upgrades: e.g. the ability to buy bigger houses with nicer views, fancy cars, expensive vacations, high-end restaurants, etc. Yet we need to make sure this opportunity is accessible to everyone with the ambition to pursue it.
On the other hand, those that choose to work less and/or don't get lucky should be able to live simple yet dignified lives. They should not be punished with existential dread because they cannot afford the necessities of life. What this often looks like today is being so rent burdened that they cannot make ends meet and fall further and further behind. The threat of homelessness and insecurity should not hang over the heads of everyone that isn't wealthy.
My first proposal is to build A LOT more housing. Specifically, multi-family housing, affordable by design (smaller, more efficient), in desirable locations, in good school districts, in walkable/bikeable neighborhoods, preferably near reliable public transit. Places people can easily live without cars (which we know are money sinks). Yes, this means building apartments in wealthy neighborhoods, so rich people accepting the poors as neighbors and in their schools. Build enough housing units and rents decline -- this would be the single most helpful outcome for those on the bottom rung of the economic ladder because housing is their largest expense. Those who work hard and smart and get lucky can still pay a premium for bigger SFH, not taking that away.
On the upward mobility side, we need to get back to some sort of free/highly subsidized state college. I'll admit, I'm more skeptical of this one because the penchant for organizations to grow fat and wasteful on a diet of "free" money. I don't want to create an incentive for people to go to college for 4 (or maybe 6) years just to have a fun experience for an unmarketable degree. The low-handing fruit are those pursuing degrees on the front lines of medicine such as doctors and nurses. We should just pay for these to address shortages, which should reduce medical costs longer term thereby helping people at all income levels. For other degrees, I'd like to see something like a feedback mechanism that informs the amount of subsidy a field of study receives. We know what degrees people earn, and we know how much they earn after they graduate. Generally speaking, higher income is a signal for value. So on average, degrees that result in higher earnings (say, 5 years out) should be more heavily subsidized. Those that have the ambition and ability would be directed to remunerative careers, and they could do so without crippling student loan debt. Whereas those that have the desire and the means to pursue an obscure unmarketable degree can still do so, but they would have to pay their own way.