However, academic recruitment is mainly application-based, and they are stuck to Equal Employment Opportunity, in which even when planning to appoint a Vice President to a vacant President position, they need to follow public call for application.
Why a department chair does not offer faculty position to exceptional scholars to attract extraordinary people?
At least for public universities in the US, you've answered your own question. Public universities cannot directly offer jobs to anyone; they must follow Equal Employment Opportunity laws, which require open applications. Even with an open search, department chairs often don't have unilateral power to offer a job to anyone. At my university, all faculty appointments must be approved by the dean of the department's college, who among other things, is supposed to verify that all EEO procedures have been followed.
But as Chris writes, less direct headhunting does still happen. For senior positions, especially endowed chairs and department heads, most applications submitted in response to the public ad are hopeless; the only strong applications come from candidates that faculty identify, contact directly, and convince to apply. (That's the explicit reason my department has a faculty recruiting committee, and not just a faculty search committee.) But even for assistant professor positions, faculty do contact promising PhD students—by email, by phone, or in person at conferences—and strongly encourage them to apply.
For even higher-level positions like deans and university presidents, universities often work with professional search firms to identify and contact promising candidates, but my impression (having worked with such a firm in one search) is that those firms are mostly good at identifying people who aren't interested. Strong candidates for those positions—the ones that are actually invited for public interviews—are almost always people that the university faculty and administration has been courting for months.
Departments do sometimes identify people they'd like to hire even before they've advertised a position. In that case, the university may create a position specifically for that person, with a very narrowly tailored job description. But then sometimes the department gets an even stronger application in response to their narrowly tailored ad, so they don't end up hiring their original target after all.