Your title question is "How to get people to trust me" but the body of your question makes it clear that you don't even understand why that should be relevant and shows that your style of leadership is based around blind obedience and fear.
It seems your question is too early; you first need to understand why it is important for your reports to trust you instead of being afraid of you and then you can work on getting them to actually trust you.
The feedback from your team shows you that they don't think you're a good enough leader to promote. Your response as giving them a stern talking to for being honest about what you can do shows them they were absolutely right.
As for a follow-up question you should ask yourself: if you reports don't trust you, are you sure they will tell you when there is a problem that you need to act on? Clearly being honest about problems earns them a "stern talking to" instead of a constructive discussion about what needs to be done. There might be a hundred things going wrong under your leadership right this moment and you would have no idea, because your reports fear you and won't confide in you because they don't trust you'll actually help them solve their problems. Which, incidentally, is your entire job. You are there to make sure your reports do their job, which is usually more "help them with their issues so they can be more efficient" than "whip them until they do what they're paid to do".
The only advice I can offer you right now is to forget everything you think you know about leading a team and reading a bunch of people management books. You've got a long journey ahead of you if you want to be a respected manager.