I'm not familiar enough with political history to confidently answer the question we appear to disagree about: Are atheist governments more or less secretive than religious ones.
I'll give it a try. Let's consider governments during the last century. The present Red Chinese government is an atheist government and certainly more secretive than the religious US government. The religious Nazi government, however, was probably as secretive as the atheist Soviets were. The major European governments are probably less religious than US ones, but I don't know if they are more or less secretive than our own. The Cubans apparently retain a strong Catholic influence, but are very secretive. South American military dictatorships were more strongly religious and have probably been more secretive than our government.
The continued application of the 1st amendment is much more important to how open the US government is than where a particular administration is on the religious spectrum.
I suggest that it was due to the higher- than- normal- for- that- time- period level of agnosticism among the founding fathers that motivated them to put in the 1st amendment. If they had all been members of the same religion, they probably would not have seen the need. If the Mormons had been in charge, for example, they wouldn't have put it in (Joseph Smith, the founder, was imprisoned for destroying a newspaper press that criticized him).
I would expect US administrations that are less religious would be more open about their activities, but I'm not sure that's what has actually happened.
Do you have any knowledge that can help answer this question?