Becoming operator for PE backed companies

I was wondering how one becomes a senior exec (CEO / CFO / COO) for a private equity backed company? Is business school a worthwhile step if the ultimate goal is to be an exec? Is spending time at a PE firm on the investing side a good use of time if the ultimate goal is to work on the operational side?

A bit of background on me. I went to a target undergrad and did two years at BB, two years at a VC fund and left to join a portfolio company on their finance team. I've been here for a little over a year and currently have a few business school apps in process. The role is great, company is growing fast and I get a ton of exposure to the senior execs who are extremely knowledgeable in the industry. The experience has convinced me that in the near to mid-term I want to continue on the operational side of things. However, longer term I would be interested in going to a VC / growth firm on the investing side or developing a relationship with some of the sponsors we work with and becoming an operating guy who gets pulled into various portfolio companies, "creates value", sells the company and moves onto the next one. I have no interest in working my way up in a F500 with the goal of ultimately becoming CFO / CEO. I much prefer the smaller teams and leaner environments of smaller companies. How do you become that "operating guy" outside of obviously successfully operating these companies?

At the fund I was previously at we had several senior execs, including my current boss who had worked for a number of our portfolio companies who we would bring in to run the companies. I was wondering if this is pretty typical in PE / Growth as well? Is business school a worthwhile use of time if something more operationally focused is the goal? Would the b-school network, which will presumably include a lot of pure investing guys be worth the time and money I would have to spend on business school? For context, returning to my current company is an option post business school.. Given my goals would it make sense to do something like working on the operating team at a PE fund with the intent of eventually moving to an operating role (i.e KKR Capstone). Are there other operational roles that would be worthwhile looking into post business school?

I know this is a long post but there doesn't seem to be a ton of information on this forum about more operationally focused roles so I thought it would be helpful to see what the finance minds on here think.