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Competition ("out-compete") implies competition on some metric such as profitability. So, if that's the metric, then they'll be profit-driven. Of course, profitability can be optimized for over different time horizons--even if speculatively. For public companies, time horizons tend to be relatively short, or at least quarter-to-quarter profitability is important.

A private company, however structured, can define out-competing however its owners want to so long as it can pay its bills and employees.

Public companies are driven by what their executives want, insofar as their owners aren't paying attention. If they were pure paperclip maximizers they would switch businesses more often, but you don't see coal plants selling fentanyl out the back.

It's not exactly what you describe, but you might be interested in looking into cooperatives. https://cdsus.coop/about/what-is-a-co-op/