We agree up to your last paragraph.

But what happens when it’s not the insurance company but rather a self-insuring non-profit religious organization? What we think of as “insurance companies” are really just “administrative services companies” for the vast majority of Americans who get health insurance through their company.

The company/organization they work for “self insures”, which means it pays/subsidizes the employees’ health costs while the “health insurance company” exists only to process claims and negotiate provider prices on the employer’s behalf.

So for a major religious organization, say, the Little Sisters of the Poor, who have a legitimate religious (not that it should matter, but I am agnostic) objection to direct funding for contraception or abortions, would you be willing to reconsider your stance?