Simple answer: it depends significantly on the liquidity of the assets being managed, and the mix of liquid (e.g. listed stocks) and illiquid assets (e.g. the unlisted equity of a wholly owned subsidiary).
The term you want to search for is "fiduciary duty
". It applies broadly, if not necessarily equally, to asset managers and corporate managers in the US (with varying application elsewhere). Those managers have discretion in how they act in order to meet the standard of fiduciary duty but this must be balanced against the risk of legal action by beneficiaries (example
hedge fund managers and the like
That's a sweeping generalization. If you mean to say, 'managers of offshore funds that are more lightly regulated than a typical US or EU mutual fund', then you are trying to pigeonhole an overly broad range of behaviors.
A key question to ask is whether a manager (either corporate or investment fund) bound by fiduciary duty is acting prudently. If you manage a company whose holdings of tradeable securities are, say, 10% of total assets, perhaps you can argue that prudence does not require you to watch those asset prices in real time. That is not the same as ignoring price movements: it seems safe to assume that Warren Buffett has at least one employee who will inform him if one of his tradeable assets has experienced a big gain or loss.
If you are fund manager ("advisor", etc.) serving US clients and you manage a fund whose NAV is principally invested in tradeable securities with daily prices, it would be much harder to argue that you were acting prudently by ignoring daily asset price changes. At a minimum it doesn't seem like something to brag about. In my opinion that is exactly why some business or fund managers highlight their aloofness: to draw attention to their different managerial style as compared to a traditional fund manager.
A hedge fund that invests in illiquid assets without a visible secondary market might not have any ability to "track" prices passively. They may have to make their own prices. Absent new information about the asset or secondary market activity, today's price is yesterday's price -- you don't need to check it.