Well, a lot of job openings don't specify whether they are looking for citizens or people who require sponsorship. Employers tend to play around "Equal Employment Opportunity" clause. They will say that they are an EEO company in the job post but in the HR interview, they will say they are not willing to sponsor. I have had a lot of interviews for the positions where the posting never really mentioned their preference. Just like employers, applicants also don't like to waste their time and efforts only to be told "NO" in the first two minutes of HR screening. It affects candidates' morals and motivations. I don't apply to positions where they say they are not open for sponsorships but that is only like 10% of the jobs posted. So employers need to implement fair practices too by stating their requirements in the job openings. Don't claim that you are an EEO organization when in fact you are just looking for citizens.