Lots of good questions. Maybe a primer on the relationship between a VC and a founder would be helpful.
Say you are a founder, you've got a killer idea, you think you can change the world with this idea but you need capital to bring it to life. Say you can demonstrate on a small scale that your idea does work, you have analysis that the market will support your idea, but you need money again to make it a reality. VCs come in, say we'll give you a boat load of cash today, in exchange for a hefty portion of your company, based on an evaluation that we're going to make. They usually take a board seat, and will be a part of your company's leadership, but remember the goal of the VC on the board may not always align with what other board members want for the company, VCs are looking to make the company's evaluation value grow so their investment is worth more than they spent, usually for a set amount of time or some sort of liquidation event occurs, be it acquisition, IPO, or shutting it down.
VCs work within a fund, a set amount of capital to invest, and that fund generally has a timeline, say 10 years, for the fund to end and realize any and all profit and losses and close the fund down. Once a company goes public, the VC can choose to do what they want, including completely selling all the shares they can convert into straight cash, or holding onto it, or devise some other strategy to distribute stocks to it's limited parters (the other groups of capital such as pension funds, college endowments, etc. etc.) for them to do what they wish.
Early/Seed, Mid-Stage, Late stage, these are generally terms to describe where a company is, and sometimes can be seen as part of the VC chain, seed / series A, Series B-C, Series D and above if additional rounds of funding are needed.
There are plenty of companies that exist to purely exist and serve their customers, by the time a company goes public, VCs are generally long gone as their funds have run their course. The VC model is a relatively new investment invention compared to industrial companies of the early 1900s in manufacturing let's say.
The pros and cons of a public vs private company can fill an entire textbook. They're just different models of operating, with different legal ramifications, different things you can and cannot do as a public company versus a private company, as well as access to capital which as a public company comes much easier (sell your share to the public) than it can when you have to pitch private investors.
In the broader economy, VCs play a small part, endowments, pension funds, other investment groups, may as part of their investment strategy partner with various VC and PE firms. But there's there's a diversity of firms that operate VC funds, it's not all just in one company (although one could argue Softbank Vision fund is just that), and each VC and PE firm has their own investment philosophy on how they stay successful.