It is not up to you to make that call. There is absolutely nothing you can do to prevent someone from entering a public facility.
Nobody is suggesting it is ok or that you should condone it. Again nothing you can do about it.
At what point did the individual in question become a coach? He is a spectator in a public facility. If by some chance Charles Manson were to be paroled, unless it violates the terms of his release he can come watch your games. Background checks are required of individuals who in the course of their duties as a coach or other team official will have contact with minor children. Have you done background checks on all of your parents? I have over the years and you might be surprised what you find.
This is certainly within your purview as the HC. It is one of many possible ways to handle the issue.
You are 100% correct.
And your answers are why I am so very pleased that convicted sex offenders have a real hard time securing employment, finding friends and fitting into a community. Our laws are great and so is the registered sex offender registry!