those employees would then pay for with their own wages
Insurance is meant to spread
risk across the entire group. Therefore if the institution made it so the insurance company allowed people to pay extra for a non-risk purchase, then unless they pay the actual cost of their personal birth control, the institution and all its employees, by definition, pay some part of that cost. Therefore the institution is using funds, probably donated in some part by members of that institution, to support a practice they are morally opposed to.
Employees likely won't opt-in when they aren't going to use it, so there won't be a pool of some people using it and some people not using it to spread the cost around. Everyone who opts-in will be pulling out of the funds.
If they make the cost the same for everyone, it's going to cost more than $30 a month for the basic pill, because some employees are going to choose the IUD, depo, nuva ring, surgery, etc. Therefore employees who only want the pill or another cheap method, or who can't afford more than $30/month won't choose to opt-in, they'll get their control from another provider. Therefore the cost will go up because only those employees choosing more expensive options would opt-in. Worse still, some forms are big one time expenses that aren't recurring. So an employee might choose to opt in for a month, or a year, have a $500-$1,500 IUD insertion, vasectomy, or tubal ligation procedure, then opt-out, thus extracting a great deal of money from the insurance without coming close to covering it.
Generally, even if an insurance company doesn't provide the benefit, employees can get the insurance bargained rate at their provider if they ask, so having an opt-in just to get the lower rate doesn't buy the employees anything either.
The only way "insurance" works when talking about regular medical costs is if everyone opts-in,
especially those that don't need or use it.
So no, it's a false position to assert that the institution and other employees would bear absolutely no cost or burden for the birth control services, even if there's an "additional cost" and especially if it's "opt-in".
If the employees are happy to bear the costs themselves, they should do so directly with their provider. Insurance doesn't work this way, and to force it breaks the model in a way that ultimately leaves the employer holding the bag.