The BSA are closely tied to government that they are quasi-governmental.  For one, they have a congressionally granted exclusive use of the phrase "Boy Scouts".  There was an attempt to create the "American Boy Scouts", but the BSA successfully sued them.

Secondly, the President is the honorary head of the BSA.  If the BSA had a "no Jews allowed" policy, can you imagine the uproar over the President being the honorary head of the organization?

From ReligiousTolerance.org:

The BSA does not have a merit badge on religious or spiritual knowledge. However, about 5% of Boy Scouts have completed a religious program with their own faith group and may be allowed to wear the corresponding religious emblem on their Scout uniform. This privilege is granted by the BSA to dozens of religions and denomination, if the course meets their standards. The emblems are supplied by the denomination, or by P.R.A.Y, an independent agency, 6 after the youth completes their faith group's course. For example, the Unitarian Universalist Association has a Love and Help course and emblem for Tiger Cubs and Cub Scouts. They have a Religion in Life course and emblem for Boy Scouts and Explorers. Permission to wear their emblems was rescinded, then restored, then re-rescinded by the BSA because the course does not meet their standards in the teaching of sexual orientation, and deity. The Roman Catholic church has courses and emblems called Light of Christ for cubs, Ad Altare Dei for Boy Scouts, and Pope Pius XII for explorers.

The BSA not only excludes gays and non-believers, but judges the religious affiliation of the scouts.

If they want to be seen as a private organization, they should divest themselves of ties to government.