The above answers focus on what's missing - why people arent open, and that exit interviews are too late.
I want to add my own experience in this. It may suggest how to change it.
I spend my work life in small to medium businesses/organisation, not large ones. Say 30 -1000 employee size. I didnt have a sense of status or distance, so I was friendly/collegial with everyone, and by nature I don't have a "fake" work face to put on. So I tend to get what non-fake often gets - more honest and on depth dialogues. Ask someone how works going, or show by your actions you want things better and don't mind rolling up your sleeves to do it and empower people... people notice that and they do respond.
Very early on, I noticed that the people who really knew things were also, every employee in the place. The receptionist and VP might have different perspectives, different slices of the cake, but they do see the cake, and both see different parts of it. Oh yes they do.
With every employee comes a free brain - use it. Your staff, junior especially, know everything that's wrong with the business. Why it demotivates. How it loses customers and profit. What the bottlenecks and inefficiencies are. Who causes friction or uses their status in a way that harms the whole, and how. Your staff smooth over a thousand issues and rough points daily, and management never know 99% of it... and why would they? The higher management above them sets goals in terms of metrics, and those goals don't include a duty on each employee to measurably improve the subjective quality and efficiency of the company's workplace. They often don't want to know things ("Just make it so") or think only their peers have insight. They may play power games, and if they don't, they might anyway, so why risk a job? They usually only want to hear some kinds of criticism, not honest total feedback, and the power dynamic influences even that. The issues were there 3 years ago, last year and now, there's no real sense of a passion and power/determination to fix them.
And that's why so much of what goes on is lip service to the idea of being the best. Because to be the best, you have to work as a team, and every last one of those things reduces your ability to all (receptionists and janitors upward) truly work as one team. Your staff would love the business to thrive and run smoothly, its not their fault if it doesn't, its a management failing. Always, 100%. Because that's managements job, totally: to ensure the quality of workplace and determine how the business works.
My preferred solution, which works a gem: be the change. Be that one manager who is those things, does those things, stands up for their team and wants to free them and empower them to be their best. Be that most unbusinesslike person - a friend - in the workplace (as far as is professional and appropriate anyway). Be someone who can say what has to be, but isn't a dick about it, and makes sure they know you really did think about their concerns and hopes too. Someone who asks casually how it's going, any issues today, and uses that as input to find the problems to fix - which your staff know but lack power to address, but you can do something about. Then come back to them with an update, next time, so they know you did something and where its at - and that if nothing happens its not because they were ignored. Take problems, including problem conduct to your staff, seriously - and any conduct that leaves them less motivated is probably problem conduct. Know their jobs in detail as well as they do, including the problems they face in their jobs. Don't focus only on appraisals for your subordinates - ask them to write upward appraisals for you... and make sure they know its just to help you be better for the team and spot anything you've got the ability to do better yourself, and won't go further without consent (if an issue needs it, you'll ask their permission first), so you'd like honesty. Do things by visiting desks as part of knowing your team, not always by memo.
Do that, spread that ethos, show that example, and you'll never need an exit interview. You'll know every day, anyway.