Surely it's a bit of backwards logic to ask for a CMS to do what an LMS does and more, because you feel LMSs aren't flexible enough?
They're catered to a task, you're not going to find an out of the box CMS that does what you want if you're unhappy with that - it'd be more practical to just make your own.
Maybe if we knew exactly what you wanted that LMSs weren't providing? I'm unsure of your setup but in my experience the whole point of an LMS was so intranets didn't have to deal with multiple CMSs.
I took a look at some LMS systems, like Moodle and Efront. I didn't like the Moodle structure. EFront has a good solution, but not the free version. I wonder to create my own LMS, but I don't have the knowledge to do this.
I thought to use a Wordpress solution because it has a several plugins that help to manage the content and to control access. But before it, I'm trying to figure out if has a better solution