It's early, so I'm mostly going to stream-of-conciousness this answer, because real estate, and real estate
policy is extremely complicated.
Tldr, the solution to housing supply issues to to build more housing. Full stop. If there was ample housing, a decision to buy vs rent would be up to the individual, and all of the talk of how to manipulate policy to allow housing to benefit group A/B/C over group D/E/F would go away.
From what I see, your concern here is about people not being able to afford to "buy a house for themselves because corporations are buying them all", and not on the "corporations that rent houses make people homeless" angle.
First, just in case, I'll address the second. Organizations that buy and rent property do not make people (as a population) homeless. Full stop. Unless they are keeping houses vacant for some other reason, if there are 100 houses, an organization buys them and rents them out, 100 families/groups still have homes. The number of housed people is the same.
In fact, you can argue that rented homes actually decrease overall homeless due to people being more willing to have roommates when they rent a property. Getting a 1 year lease with someone you know through a friend of a friend? Totally reasonable. Going halvies on a house with someone you know through a friend of a friend? Absolutely terrible financial decision. I highly suspect that people who own their home are overall less likely to actively go out searching for a roommate (I wish I had a reference on this)
Overall point, if the goal is to reduce homelessness, you need to build more houses. An organization/corporation renting units does not decrease the available supply. Arguments like "Well, organizations do slow remodels that take units off the market. If an owner owned it, they would live in it during a remodel!". The number of houses that affects is so small it's hardly even mentionable.
To the first point, many people are arguing that the way to do this is to make it so that you can't "profit" on owning a home. In the US, owning property is seen as the path to financial freedom. Enacting policy to actively prevent that will drive many people away from even wanting to buy. If there is artificially limited finanicial upside in buying a property, the math has changed and a lot of people may choose to rent instead. But the policy changes have made it way more difficult to find anywhere to rent. You're stuck buying something, being fully responsible for all of the components of it, and then hoping you can sell it when you need to. Realtors love this one trick to force people to buy/sell homes.
Finally, don't forget that the US already subsidizes home ownership to a large degress in the US. Federally backed mortgages that keep rates on 30-yr fixes terms well below a market rate, FHA programs so people can buy a home with 3.5% down (ie, leveraging thsmselves ~30x), property taxes and interest being tax deductible, 250-500k in tax-free capital gains when you sell a primary residence, and being able to use any improvements to increase your tax basis. None of these benefits exist for renters.
As discussed earlier, the main issue with housing is supply. When you have limited supply, and programs that are designed to increase demand, prices are going to go up. Whether there are startups/corporations involved or not.