So your concern is rather straight forward. While you do not intend to hold a bias against someone of opposing beliefs you recognize despite consciously feeling no bias subconsciously that bias is still very real.
Objective Hiring process
Even taking the EEOC, ethics, and morals off the table Objective hiring is very important. If you don't go out of your way to ensure objectivity then the choice of who to hire becomes primarily subjective. Subjective hiring is when people make mistakes and hire the underqualified charismatic person, allow personal prejudices and biases to bleed through.
How you make your hiring process objective isn't as important as doing so. A flawed process is better than no process.
In my case the first step is I put up an application process where applicants need to apply online. This formats their resume to be exactly the same as all resumes we receive. (We allow them to attach their own, but to be entirely honest HR only looks at those far enough to see if they're well made or have any obvious red flags)
When I'm ready to go through the resumes I print them out, as they go through our application I don't get names or contact information. Just an applicant number, working experience, educational background, objective, etc. This really helps to vet the worst bias such as religion, gender, age, ethnicity, etc. You can still have bias regarding choice of education, past employers, etc. but honestly removing that information makes the resume completely worthless in my opinion as not all employers and schools are remotely equal.
from here we review the resumes in a "Beats Expectations", "Meets Expectations", "Does not meet expectations" The final pile is removed from consideration.
Now we arrange phone interviews based on our favorite resumes working our way down until we wind up with ~Five people we decide to ultimately interview in person.
On the phone we just probe your personality and technical experience. This is honestly mostly just to watch for red flags of people who won't fit our needs. (most of the time this boils down to bad attitude or clearly not as knowledgeable as we need)
Then we perform an in person interview where the person meets several members of the team after which we all come together to decide what we think and we end up hiring whomever the team collectively sees as our best candidate.
It's not a flawless system, but by keeping the first phase anonymous then involving the entire team in the process it really helps avoid biases as well as by involving the team it helps avoid hiring clashing personalities. (and sometimes you'll have a team members who have outside knowledge about candidates that can be very valuable in avoiding bad hires and promoting a good hire you might otherwise overlook)