Question

What belief is built on the first amendment's guarantees that the government will not choose an official religion or prevent citizens from practicing any religion they choose?

Answer 1

The separation of church and state - APEX

Answer 2

The belief is that The separation of church and state.
Including certain religious organizations within the governmental process will indicate that the government would always work in favor of the specific religion This would be really contradictory to the first amendment that guarantee EQUAL treatment and freedom for ALL  religion.