I think your test ideas are relatively good. But what would prevent a creep from getting through the screening? Also overpopulation isn't a thing. And passing a test doesn't mean you are good at the job.
http://www.firetown.com/2010/12/03/the-overpopulation-myth-the-entire-world-population-could-fit-into-the-state-of-texas/
It wouldn't be one written test though. There would be extensive psychological tests among other things. If it was created properly there woulc/could be a high chance of preventing offenders from access.
Over population is very much a thing in cities. Just because that person claims we can all fit into texas back in 2010 doesn't mean we can or should going into the future.