Not sure of this will be helpful or if it's even wise of me to enter the fray, but here goes...
1. Software terms don't travel. One man's Project is another's library is another's vault & etc. Don't assume that a Library or project or archive or anything is the same in Resolve as in FCP or Premiere.
2. A Library in Resolve is an SQL database. They've been around for a very long time, are a very robust and efficient means of storing data. Lots and lots of data. You can have multiple Libraries, for example by year, or by type, or if you're a larger facility by client. You can put the library anywhere you want.
3. A Library contains Projects, as many as you want, each may be sorted and organized however you want. Projects always live inside the Library (e.g. the database). So if you have a Library named Netflix (your client) you may have Projects for After Maria, Young Wallander, Dead End, The Sinner. They may be organized into folders, or seasons, etc. Or you could make a library for each show and a project for each season. Many ways to organize things. When you are dealing with hundreds of projects, flexibility and organizational efficiency are good things to have.
4. A Project contains project settings, links/pointers to your media and other assets, and Timelines. All of these are (as far as I can tell) database records (remember the Library..?) and essentially pointers to the actual assets and resources (e.g. plugins & settings) being used. There's no media, no assets, nothing else there, just records that describe the project, it's assets and structure.
5. Since the media & other assets and the project files are completely separate entities, you can store a Library (database) anywhere (NAS, Shared drive, the new Blackmagic Cloud, etc) as it's relatively lightweight and efficient, while your media files can be on high-capacity high-bandwidth storage (SAN, etc). Separating the two allows for more flexibility, especially in a commercial multi-user environment, which is where Resolve originated (as did Fairlight).
6. Resolve was and is (disclaimer - my understanding and opinion only here) aimed at commercial/enterprise users, typically post-production facilities, broadcasters and the like. The fact that they made a free version doesn't change their core market, it only allows more people who are curious about it to check it out. If I was doing TikTok videos on my iPhone, just because I could borrow an Arri or Red camera for free to shoot with doesn't mean I'm going to do better work or find it an easier experience, nor would I expect those companies to cater to my needs and change their product accordingly, despite there being millions of TikTok creators, compared to a far smaller number of professional DPs shooting commercial content. Different tools for different markets/audiences.
7. There are different types of Libraries, Local, Network, and Cloud, but I don't want to make this more complicated than it needs to be.
8. I'm not sure Resolve is made to be intuitive, I think it's designed to be extremely efficient in fast paced environments and is focused on effective workflow management. As with many professional tools, it requires training and knowledge to get the best out of it - experienced operators. It's not an "anyone can do it" kind of thing, its a "professional's tool" one. That may be why they offer (free) courses to learn the platform, or parts of it. As a personal aside, I wouldn't call ProTools or Reaper intuitive either, until you've used them for a while and understand how they organize their workflow (I'm an audio guy, thus the audio references).
9. Yes the manual is extremely long, I find using the search function usually turns up a few pages outlining exactly what I'm looking for. Based on my own experience, I would suggest investing the time in either the free online tutorials/courses on the BMD website, or signing up for one of the (also free) online courses they hold several times a year. I know the expectation is that once you know one system, you should be able to easily move to another, but from my own experience transitioning through several DAWs in my career, that's often not the case. Especially for more experienced users who have established their own way of working, when the new system really calls for a different one. It can be a pretty big transition!
Page 71 of the manual describes Libraries, what they are and how they work, as well as how to manage projects within them.
Page 67 walks through projects and project management (within a Library, or database as they used to be called)
Page 81 explains how to use Dynamic Project Switching.
Page 3951 gives a more in-depth look at Libraries, Projects and the like, which I've tried to summarize here.
(All references in regards to version 18 of the software).
I hope this can be of some help.
Cheers,
Thor