There are several things that I find wrong with your ideas.
Firstly, there are other people and agencies besides the government searching for alien signals. SETI, for example, is a non-government organisation. There is far too much to be made, in terms of glory, becoming a historic figure, and the like, for these searchers to not reveal any evidence they have.
SETI has released evidence to support ET's before, too. SETI found a signal that repeated every 1.33 seconds, and initially stated it was evidence for an ET. Although this was later found to be a pulsar, SETI can hardly be blamed for mistaking this unknown phenomena that contradicted so much of what we knew previously about celestial objects. The "Wow!" signal is a good argument for the existence of ET's, and the discovery of that was officially released to the media, too.
A news article I read (forget where I read it) said that the time between the discovery of an unknown signal, that could potentially be from an ET, to the news being leaked world-wide, was less than twelve hours, despite any efforts to keep it secret while trying to verify it's accuracy.
Secondly, this supposes that every single person that has knowledge of the existence of ET's with radical religious ideas supports religion either directly or indirectly, yet has no trouble, in terms of religious morality, in covering up about the situation. Certainly, it is possible, I would consider it highly unlikely.
Still, interesting idea.