My bet is the warning that the public can’t handle it is more a fear tactic to deter investigation. Assuming the speculation of compartmentalized programs contracted with private companies, there is a lot of financial motives along with potential illegal activity. The groups involved wouldn’t want the history of how they’ve operated over the years to come to light. And, most importantly, they want to maintain control over any potential knowledge or technology obtained. We’re talking about potential of 70+ years of information withheld from the public and being used by private companies for their own gains.
There would also be some level of change of public perception along the lines of what you suggest, but I think it’d start with focus on the government and the individual companies involved. I doubt there would be a lot of immediate shifts in the public perception. There will always be a large portion of the population that would ignore or deny any disclosure, so even the most jarring disclosure would take time to ingest.