What do HR employees do, when a potential employee's personal references do not respond to their inquiries?

It will depend on the company, its policies and the human resource person.
Often, if there is another person also being considered, they may just move on to that person.

They may call on phone and confirm.

They may call the applicant and ask them to assist.

Then one would almost begin to believe, that if an employer advertises to fill a position, and needs help quickly... an applicant, having an application already on file, with these checks completed already, would have a substantial advantage? (Assuming they qualify)

Sub-question: I am going to pay my personal references, to make the time to prioritize their responses regarding me. So... How would one go about implementing this same idea, regarding past employers?

Call up the HR or manager at that location and tell them, to expect a reference check. You may even ask their policy. When I worked for one health care firm, they would only confirm dates of employment, nothing else

Personal references aren't given much weight. Usually such a person says all sorts of wonderful things about you that can easily be untrue. (Paying people for this is throwing money away.)

There is nothing you should do regarding 'implementing' a plan with past employers. Can you try to bribe them? Of course. Could easily be the worst move of your life.

I never paid much attention to personal references, only professional work references

In my companies and work, never even contacted or used personal references

Nobody is going to give a personal reference that will bad mouth them. So personal references aren't given much weight. However, if they don't respond, that may have influence. Past employers may carry more weight.

Its rare that resumes are kept on file for very long. So don't expect HR departments to go over previous applicants.