No, this is incorrect. It doesn't matter who owns the stock as long as someone does. It's irrelevant if one person owns the stock or it's distributed equally among workers (or something in between.) The incentives are the same. In fact, distributing stock more widely could lead to better decision making by crowdsourcing wisdom.
But for now the stocks are mostly owned by rich people who want to get richer not matter what and it seems to me that the whole market is so used to that, that it would take a radical system change to make it happen without the economy as we know it crashing hard.
I don't know that it needs to be an instantaneous change. Something gradual could work, at least in theory. On the other hand, I think we need to do something not just for reasons of justice but rather we need to internalize a lot of these externalities just for survival, which wider ownership would naturally do.