I am actually not sure about the coroprate home ownership point. Here in Germany renting is much more common and accepted compared to the US, and i think there are lots of situations where this makes sense. However both in the US and here in Germany the systems need changes. And i think they should mostly target land ownership rather than the houses themself. What drives up the prices in desired areas are mostly increases in land value, not that building houses got that much more expensive (although that is also a factor).
And most of that value gain are from external factors rather than the owners own merit. If someone builds an architectually great and energy efficient house or develops land, then it is fine if he gains value from it. But if simply owning the property improves the value over time, because society around it builds nice schools, parks and so on. Then the owner hasn't done anything and that profit should be taxed completely away. If that makes sense.
That said there probably should also be a mechanism to support the first home people own to counteract scale efficiencies that corporations might be able to leverage.
Not sure if outright banning stocks for politicians is the way to go, but there should be more points regarding transparancy and conflicts of interest. Also not just during their time in office, but after that aswell.
I'd have no issue with politicians holding a borad market index fund.