Problem if Shareholder is a Director
There is a potential conflict of interest (not only for a chairman but also for any director or employee who owns shares in the company) arising from the possibility of insider trading.  Suppose the chairman of a quoted company has a large shareholding and, through their position, becomes aware of information, not yet disclosed to the market, that means the company's prospects are not as good as previously believed.  They might then want to sell shares immediately, rather than waiting until the information has been disclosed and as a result the share price has fallen.

Since directors including chairmen are supposed to act in the interests of the company and its body of shareholders, there are two possible conflicts here.  Firstly, by focussing on whether to sell their own shares and arrangements for doing so, a chairman might neglect their duties to the company at a critical time for it.  Secondly, the selling of a large block of shares would tend to depress the share price to the detriment of other shareholders.

This is not to say that a chairman should not hold shares. A relatively small shareholding would be unlikely to give rise to a serious conflict of interest. In many countries (see here) the issue has to some extent been addressed by making insider trading illegal (although the detailed rules differ between countries).