would that in turn mean only one person could own a company and not allow investments?

you can already own a company with multiple people without the stock market.

would that involve getting rid of publicly traded/owned companies?

you can already have a public company, that is not listed on a stockmarket

Then what’s the alternative?

any of those that already exist.

I guess it would also be possible to image some new implementation of public trading, one that does not enable corruption and white-color-crime so much as the current one. Maybe something with more transparency and public oversight. Keep in mind, that the stock market in the US was implemented by already rich people, who had no incentives to make it fair. All fairness was implemented later as an after thought.