You're not wrong, but shareholders look at their investment very differently than stockholders. Private shareholders can't necessarily cash out whenever they want because the sale of private equity is usually tightly controlled by the company. This means they need to be interested in long-term growth and success. While public stockholders can also hold their shares for a long time, there's much more ability and incentive to buy and sell quickly to make a quick profit.

Anecdotally, I worked for a publicly traded company for 6 years before they got bought and taken private by a private equity group. The way profitability and trends are measured is night and day. As a public company, everything was hyper focused on quarter by quarter results. One underperforming quarter meant a tank in stock prices, hiring freezes, and a general sentiment to the employees of "quit spending money on expenses if you want to have a job next quarter". Being controlled by private equity, they're most concerned with year over year growth and the long-term stability of our operations.