From a Historical perspective, Recruiting has certainly been around for thousands of years, and, expectationally, will definitely continue to exist for a long, long time (at least if Humanity continues to exist).  

But, will "Headhunting" continue to exist?  The mere word evokes images of primitive savages with curare-tipped darts hiding in the jungles of the Amazon or New Guinea, intent on predation of fellow Humans...  

Yet, it was perhaps the World's greatest Executive Recruiter, Gerard Roche, Chairman of the Board of Heidrick and Struggles, Inc. (the Search firm that placed the CEO of Google, and charged a fee of $100 million), who said: "'Headhunting' is a great term for it -- because we really do look for the best, and the brightest."  Far from being a term of derision, Roche converted the moniker "Headhunter" into an automatic accolade.

Still, let us return to the question of the persistence of this Industry called "Headhunting":

Let's look at the definitions:

re·cruitrəˈkro͞ot/verb gerund or present participle: recruiting

1. enlist (someone) in the armed forces."they recruit their toughest soldiers from the desert tribes"
   • form (an army or other force) by enlisting new people."a basis for recruiting an army"
   • enroll (someone) as a member or worker in an organization or as a supporter of a cause."there are plans to recruit more staff later this year"

 ------------------------------------------------------------------------------------------------------

"Headhunting," on the other hand, was basically invented in the mid-1920's, as a profession.  We are nearing our 100th year as a field of professional endeavor, but we can't be safe to become complacent, yet. 

head·hunt·erˈhedˌhən(t)ər/noun

1. a person who identifies and approaches suitable candidates employed elsewhere to fill business positions."a headhunter offering you a wonderful new position at a higher salary"

-------------------------------------------------------------------------------------------------------

With the advent of Artificial Intelligence and "Social Media", it does make sense to have concerns about whether the relatively "young" profession of Headhunting could be done instead by automation, more efficiently, saving time and money, by computer proxy. Yet, the presence of "Human Factors" is always important, as well (https://en.wikipedia.org/wiki/Richard_Pew)

I learned this (eventually) because my family was friendly with the brother of Richard Pew for many years.  Richard and John Pew (Richard's brother) both attended Cornell University, like my father and grandfather did.  Richard Pew was an American Épée fencer who finished 4th in the Melbourne Olympics in the 1950's; higher than any other American fencer, in any weapon, for 24 years. This was a record that stood for 28 years, set by a mathematician and engineer from Cornell University; not surpassed until Peter Westbrook won a Bronze in Men's Sabre in the 1984 Olympics in Los Angeles. 

Most recently, however, it was my club-mate and fellow Princeton-grad friend Soren Thompson who led the US Men's Épée  Team to the World Championship in Paris just a few years ago...  This was a defeat of the French Team that had been World Champions 8 years in a row.  This was an unprecedented event in U.S. Men's fencing, although Mariel Zagunis had previously triumphed as World Champion in Women's Sabre (an event that didn't even exist, officially, when I was first learning how to fence).

Soren and I shared the same coach (Michael D'Asaro Sr.) and bouted together multiple times.  I was never able to beat him.  Still, I very much doubt that any of this U.S. Fencing History would have happened without the pioneering efforts of Richard Pew, who was also the President of the Human Factors and Ergonomics Society (https://hfes.org).

The point that I am making, simply, is that "Human Factors" cannot be ignored or replaced that easily.  Even with perfect Artificial Intelligence, Humans are still needed to deal with the candidates identified, because of the enormous complexities of the possible permutations of possible situations that can arise. No "program" can deal with all the eventualities involved in the process of Headhunting and hiring. 

"Fencing robots" do exist, and they are formidable, but the physical complexities of programming a fully-autonomous robot to fence (within the rules, of course!) is far, far simpler than programming an automated Headhunter/Recruiter.  This may happen within my lifetime, but I doubt that the precious "Human Factors" elements can ever be replaced.