Not only will AI run the company, it will run the world. Remember: a product/service only costs money because somewhere down the assembly line or in some office, there are human workers who need to feed their family. If AI can help gradually reduce human involvement to 0, with good market competition (AI can help with this too - if AI can be capable CEOs, starting your business will be insanely easy,) and we’ll get near absolute abundance. Then humanity will be basically printing any product & service on demand at 0 cost like how we print money today.
I wouldn’t even worry about unequal distribution of wealth, because with absolute abundance, any piece of the pie is an infinitely large pie. Still think the world isn’t perfect in that future? Just one prompt, and the robot army will do whatever it takes to fix it for you.