There's also this to consider: if the 1% own all the industries and they're extracting > 50% of the labor-value of each worker who works there, then for every job lost, the 1% lose more income for that than the 99% do. They gain wealth by selling products to other people. If those sales dry up, e.g. in the no-jobs future, then there's no real reason to believe that further wealth concentration is all that inevitable.
Just because its in the best long term interests of the rich as a whole doesn't mean that its something they will actually do.
For instance is it the long term best interests of everybody for toxic chemicals not to be dumped into our lakes and rivers, but if industry gets its way stuff like that will be legal. And that's only stuff thats 100% clear is terrible for everyone including them.
On things that are less clear its highly questionable if they will even agree with you, much less actually do it when there is a direct cost to them personally.
To actually do these programs the government will need lots of money, and since the vast majority of people will be poor it would have to get it from the rich. Thus, it would be in the best long term interests of the rich in this scenario for them all to not only A) Work to pay a progressive amount of the taxes as a class, but also B) Actually pay the amount they are supposed to as individuals instead of using loopholes to wriggle out of it.
From looking at things historically though that isn't actually something the rich ever actually agree to do willingly. Now, people in democracies might get tired of all being dirt poor and vote for high taxes on the rich and corporations and use that money to give themselves things like healthcare and a universal income
e.g. say the 1% own 90% of the farms. But it's clear they don't eat 90% of the food. e.g. ownership and consumption are completely different things. The 1%'s wealth is mainly paper money, not measured in terms of raw resource consumption. Average people in fact consume most of the resources, and wealth is built on top of resource-delivery systems for regular people. e.g. if farms become automated, food will become much cheaper to produce (that's why they automate in the first place). But hey, everyone lost their factory jobs, how are they going to afford food? But remember, food's much cheaper to produce now. e.g. a food stamps equivalent program would be much cheaper to run, too. e.g. there's no real logic in the idea that incessant productivity increases which is really what we're talking about are going to drive most of humanity to near-extinction levels of poverty. e.g. there are already programs like food stamps and medicaid, and if everyone's out of a job because everything is automated, the cost of running those programs also plummets along with the cost of everything else. Catastrophe averted.
Food is not going to be cheaper in the future. Global warming is going to cause cause massive crop die offs from changing climate systems (mainly massive droughts and heat waves, although massive floods will also have an impact) which will decrease supply. Demand is also going to increase as the population does as well.
So even if costs to produce it do go down decreased supply and increased demand will more then offset it.