Feedback needs to be anonymous or it will always be worthless. Few people have the courage to point out even blatant truths to their employer. Just set up an email that anyone can submit to anonymously.
I think this significantly undervalues the importance of developing a culture where things can be talked about in a frank and open manner. It may be rare, but is extraordinarily valuable, and if people are never given the chance to have uncomfortable conversations, then there's no possibility of it developing.
When anonymous feedback comes in and its taken and acted on positively the team may be more likely offer up non-anonymous suggestions, a foot in the door if you will. I may be totally off base though I don't have that type of real world experience.
This is basically how I feel about it. If my employer acted positively on some anonymous feedback at a first, I feel like I'd be a lot more likely to feel open and willing to engage knowing there's good faith. Unfortunately this doesn't scale to larger companies, though. What the OP is describing is the classic problem of management.
Another commenter pointed out that company culture is most likely the underlying issue with people not being honest during their post-hire interviews. If that's the case, then the culture is already damaged.
Maybe they could begin to repair the culture with the suggestion box, by asking why everyone is so hesitant to critique during face-to-face interviews. That alone should show that they're making an effort, and acting on any of the suggestions will further the cause. Basically, use the anonymous suggestion box to get rid of the anonymous suggestion box.
It's not cultural; it's human. You can't undo it. People who believe their culture is the exception that causes employees to be honest with their superiors are naive.
What generally happens with strong technical employees is this. While they're idealistic and new, they will be relatively honest and direct (simultaneously trying to be polite) for the first couple of years. After a while, they will try to figure out why they've been forced out from their company every 6 months.
It will click that maybe the boss didn't really mean it when he asked for an inventory of problems and concerns. Maybe all those people who smiled and nodded while a technical "debate" was going on were secretly developing resentment. Maybe people disliked them because their behaviors were interpreted as snobby, arrogant, insensitive, condescending, or detached, despite the politeness with which they delivered their opinions.
Then, they will either a) nope out of the career track entirely and become a consultant/freelancer/entrepreneur; b) lie to themselves and believe they can find the magical land where this doesn't happen, which only sets up for a harder fall down the line when you realize no such magical place exists, because people are people everywhere; or c) embrace the realities of human collaboration and try to raise a successful career from the ashes.
I've done both A and B. A usually loops back around to a full-time gig at some point. Some people will remain unaware and try to believe in B for the duration of their career. When you exit denial from B, you have to embrace C (even if you loop back to A, C is informative for your ongoing ventures).