Privately held companies issue shares as well, not just publicly traded ones. Shares just represent ownership interest in the company. If you and I formed a company and jointly invested equally, we would each own 50% of shares, or whatever our articles of incorporation and bylaws set forth.
A conservatorship is not just for health related issues, though it can be. Someone can be appointed by you or by the court as your medical proxy or hold your durable power of attorney for health care. A conservatorship can really be designed with any contours the parties and the judge want. There are of course financial conservators appointed to preserve the assets of the person under conservatorship (the subject of the conservatorship cannot make business decisions, must seek permission for investments or spending, etc. They basically have no decision making power and have to ask permission), and there are personal conservatorships (like the one Britney Spears is under). Under a personal conservatorship, the conservator can make decisions like where the subject of the conservatorship lives, whether she is enrolled in school, if she is allowed to marry, etc. Conservatorships can be designed to cover all rights and responsibilities a person would otherwise have, not just to be "on guard" in case someone's medical issues flare up. (So, Britney Spears residency in Vegas was a deal brokered and entered into by her father, if she wants to marry her boyfriend, she has to have permission, and he has decision making power over what doctors she sees and what her treatment regimen is.)