I think a classic super-star COO example would be Sheryl Sandberg, who was COO of Facebook/Meta.
She was practically running the company. Zuckerberg mainly focused on strategy, vision while she made Facebook a well oiled profit machine.
The role responsibilities are nebulous and change based on the company, but can generally be described as "whatever is necessary for the business to run". That could mean they focus on running Sales, Marketing, Support, Customer Success, but they could also oversee Operations and in some cases Product.