Absolutely spot on. Executive search firms and recruitment companies find candidates for jobs - they do not find jobs for candidates. 

I  do  apologise  if  I  am  just  stating  the obvious, but  trust  me,  most  people  have no clue about how executive search firms work.  And  why  should  they.  You  can imagine the surprise and increased frustration  that  grew  inside  me,  when  I was on the other side of the table (some 15-20 years ago before joining the world of executive search). No headhunter wanted to talk  to  me  when  I  called,  they  didn’t return calls when I left messages with an assistant or on voice mail, nobody bothered  to answer emails either. Aaarrrgghhh.

And spot on regarding the second part, that anyone can get into the recruitment business. Every Tom, Dick and Harry call themselves a headhunter nowadays, even all they do is shopping on the internet and in their database. It's the kind of small agencies who invite you to a hotel lobby or Starbucks because they operate from home or a small office space. Many are not even registered with the Ministry of Labour as the Thai law requires. Oh well.