The teams I've been on that worked the best, took initiative, and didn't need the boss around, were teams where the manager was willing to roll up his sleeves and do the work... and could actually do it without making things worse.
I'll give a few examples.
I started out in ops and we'd have to answer the phone. The boss wanted the phone answered on the first ring. Rather the beating us over the head over and over again, if he was there and the phone was finishing it's first ring, he'd pick it up and field the call. We'd all feel like shit, and pick up the phone faster. And we saw that he wasn't asking us to do anything that he wasn't willing to do himself... but he gave us the chance to do what he wanted before he would step in.
In another case I had a manager who would regularly ask me to do all kinds of off the wall stuff, but he was also doing it. If he could do it, he'd make the attempt, and in some cases we'd compare and contrast our solutions/ideas. It was collaborative, rather than him simply barking orders at me.
Most recently, I have a boss (boss's boss actually) who started out as a very authoritative boss. He said jump and he expected everyone to not even ask how high, but to jump higher than previously thought possible. He was always asking for this report or that and everyone was really annoyed by it, and him in general. Then he pulled me in to ask a question, and it wasn't about me doing something, he made his own report and was asking how to share it out, as it wasn't obvious. He was asking how to fish. That made me respect him a little more, and over time that could build into me wanting to go the extra mile (but it will take time after the rocky start he had). Trust is hard earned and easily lost; I'm not going to break my neck for someone I don't trust.
With all this in mind, they still need to trust the team. Trust is a two way street. If a team is being micromanaged, they won't do anything on their own. They'll wait for direction, because the exception is that if they make their own decision, someone will overrule them and they'll have to do double the work to "fix" it. Autonomy is a big deal. If they feel they have a hand in the solution and are building something they feel like they want to build, instead of something they are told to build (that they secretly think sucks), they'll probably be much more interested in the product and be excited to release it. I like bringing my ideas to life, but will drag my feet all day long on building something I expect to fail.
Nothing kills my sense of urgency more than not respecting the person I report to, and I have a hard time respecting a boss who simply points and yells. If a boss ever tells me to do something, "because I said so," without being able to explain why... I'm done with them. If something is truly urgent, explain to me why it's urgent. We're all adults. If you want them to go faster for the sake of going faster, why? Going faster generally means cutting corners. Why do those corner need to be cut? When looking at the trade offs, what trade off should they be making? Those choices can't be properly made when the engineers are in the dark.
I had a sense of urgency, because I worked on keeping production running for a whole data center with a small team. If I was slacking, I could see others needing to pick up the slack (including my boss), so the culture reinforced the sense of urgency and it was clear why it was important. If it's unclear why they need to go faster, and the culture doesn't support it, and it's simply their boss looking for a bigger stick to whack them with... they're not going to respond to that.
People also want some recognition. When I was running our team, we did demos at the end of every sprint. And every sprint we had almost the whole team looking to demo something, and they would bend over backwards to try and get whatever they were working on done for that demo. I would actively tell them to do less, plan for less, go slower... they were pushing to do more and go faster. They were doing work they cared about, in a way that they believed was right, and were excited to show it off for other people to see and hopefully use. I was asking almost nothing of them, but they also knew me and that I was ready and willing to do everything they were doing... because that's exactly what I did for many years. They'd always ask how we could get more people to the demos, and I asked around to other very prominent teams, and we were getting more people showing up to our demos than the model team that was put up on a pedestal for the whole org. We also rehearsed the demos many times to get them tight. This was our face to the rest of the company, and we wanted to show up well and make everyone look good. I think they appreciated that a lot. They had a chance to show off, and I tried to make sure when they did, it was well received. We had several people say our demos were the best they had ever seen at the company. This makes everyone on the team look good and want to work hard to be part of the next one. That was never my plan or objective starting out, but that's what ended up happening.
When we were preparing the demos, they followed my lead. I put the framework together for them to slot their stuff into. As we were putting the demo material together, if something was off by one pixel, I'd call it out. It's one of those things that doesn't sound like a big deal, but when people are watching something just feels _off_, even if they don't know what it is. I felt like I was being a bit of a stickler, so kind of apologized for nit picking as I said it, but once they saw it too they wanted to fix it. Pretty soon after that, I didn't need to call these things out, they started to notice these little things and call them out to each other. You have to set the bar, not with your words, but with your actions. I was also big on metrics to show the impact our work was having. This was a hard one, but I think I eventually got them to care about that as well. We had it in all our demos, and I tried to get them to get these numbers up front so we could better pick the work we did to have more impact. I ended up pulling metics first, and showed them my process. It took a good while for them to come around and see the value, but eventually, as with the one pixel thing, I could mostly sit back and the team would start to police the rest of the team to make sure they had their metrics, and looking at different ways to show it to get the biggest impact. Again, I placed urgency on it and was willing to make it a point of urgency for myself, and the rest of the team eventually followed.
It sounds like you're starting off from not a great place, so it will take time to win a team over. If you change overnight, they won't believe it's real. You'll have to be consistent until they slowly start to trust you. For that to happen it needs to also be real, and not some act where you're only doing it to squeeze every drop of productivity out of them. If that's what's going on, you'll slip up, they'll see it.. and it's over.