True! But if Catholic organizations did not have government interference to begin with, they would have only hired Catholics.
That is not true. Passing reasonable laws that prevent discrimination is not "government interference." Furthermore, as already explained, even if they hired only Catholics, that does not give the EMPLOYER the right to made a decision for the EMPLOYEE that will force the employee to adhere to a religious teaching that they might not choose to adhere to.
The decision on whether to use birth control or have an abortion is not in question. Does the employee have the right to force the employer to pay for the birth control and abortion? That's the issue and I hope the Church never lets the government forget it.
Force the employer to pay for birth control? That's not the issue. That's not happening. That's not going to happen. What's going to happen is that the employer will be required to provide health coverage plans that will include birth control. The employee will ultimately be the one paying for it. They will be the one who chooses their plan, they will have to pay the difference in premiums, they will have to pay the co-pay and deductible costs for the doctor's visits and prescriptions.