We're in the knowledge worker age now. If you're hiring "hands" then hiring someone who has done the work before is good. Your example of a cleaning person would be hiring hands.
But, today's knowledge worker must be able pick up new skills on a daily basis. Past behavior doesn't predict anything. Today's problems are different than yesterdays. For example, in the technology field hiring based on what a person did yesterday is gonna' get you old technology. Potential is the only thing that matters now.
Resumes are for covering the HR department's ass. Anyone can produce a resume that makes them perfect for any job. Interviewing is the only way to know if someone is right for the company.
If your example is your actual situation, then you're not facing a recruiting problem its something completely different. I can tell you what the problem is, but it's probably best if you figure it out yourself.
My background is on Human Resources and I am quite familiar with the STAR method. However, we do apply it differently from how you have described above. This method is usually used to determine how an applicant thinks and would handle a situation.
You give him a sample Situation and let him respond. From his response, check that he has figured out the correct Tasks that needs to be done, what Actions has he taken and finally, check what the Results of his actions were.
This simple exercise will let you know how an applicant thinks and how he would address certain situations.