One experienced CEO I worked with was great at delegating AND empowering key people. He knew (and often told us) this was key to his ability to grow the company into a true enterprise. I was promoted twice in the 2 years I was there. I was employee #30 when I joined, since I'm still in touch with the CEO, I learned they just hit 350 employees. (This is a privately owned, non-VC-backed company).
One inexperienced CEO I worked with was all about delegating tasks (I suspect largely because he didn't want to do any of this work, and/or felt his employees would be better at it than him) but often struggled with empowering his people to make decisions to get done what they need. He tried though, and I suspect will be better at this. This was also partly because he was simultaneously running experiments on how to build a company with a novel style of management - and we were working together during the very early stages of this.
Another inexperienced CEO I worked for had very little ability to entrust anyone at the company to do anything, let alone make any decisions. This was all delegation, zero empowerment, and zero trust. It was a complete hindrance to being successful, and will be impossible for that company to grow very much in that style.
The successful instances were actually very early in my career; the unsuccessful much later.
The problem with many startup founders is they really have limited experience to work off of, and not yet a lot of confidence in what they are doing. It takes a lot of confidence in yourself, the person, and your business, (or faith!) to hand someone the reigns on something important and not look over their shoulder every 2 seconds.
So first, your expectations for what you want to take over need to be realistic. The founder(s) need to be on board with giving up this responsibility and it needs to fit into the role they envision you fulfilling. Also, this should go unsaid, but you need to actually believe you can do it. If you still have doubts, that's ok, we all do; but before you try convincing anyone to give you this responsibility, you should have some voice somewhere inside of you telling you that you could probably do this, and you truly want to.
I hope this is helpful