It won't be generic "staff" but a trained HCP / pharmasist who is doing the dispensing. (The UK has similar specific exception in abortion legislation). The right to conscious objection is an important principle when it comes to employer/employee relations and society in general.
The dilema of just because we can, should we is the basis of a lot of specific rights in the US Consitution. The States were themselves formed by various groups who wanted to create their own rules rather than follow existing ones.
Faith pratice has first amendment consitutuonal protection in the USA so Walgreen are respecting their staff's right to work and have a belief. The problem is if Walgreen management don't rank the service provision as a right for the customer to have a different belief.
If the employee needs reasonable adjustment and the employer decided to accommodate the adjustment, rather than remove the employee on grounds of being unable to preform the role, it is the responsibility of the employer to ensure that that part of the role is transferred to another employee.
Long way to say they need to ensure that they employ 1 staff member per shift to dispense birth control.
The same conflict arises when the HCP is providing care. Either sex can provide personal care but should the employer accomodate the HCP without taking account of the individual receiving care.
In the US positive discrimination legislation, has never been "popular" from maternity to collage places. Even the idea (post covid) that all should have a right of access to basic health care is not popular. Health care (premium payments) as an employment benefit provision is a high cost which can vary by employee.
Insurance companies are allowed to health check an individual and then adjust (charge more for) the premium.