Why are there so many intermediate-senior UX job openings out there, but only a handful of entry-level opportunities? What are these "experience qualities" and how do I learn them?

Unfortunately the short answer is... many of traits employers look for in experienced talent are things are really hard to learn by yourself because it is all about the "interaction" with other people. It is simply impossible to do UX by yourself in a vacuum.

That is not to say you can't gain these skills if you can't find a job in UX. The most straightforward way to work on projects with developers, product managers, project managers, UI designers, business stakeholders etc. is to find employment as a UX designer. As you work with a team to learn and try different things, you gain experience and judgment which will help you make decisions.

Okay, so that is pretty vague... you just get experience from working.. care to elaborate?

In my opinion, some areas where experience sets a designer apart is being able to judge:

1. When to ask more questions and how to best ask it
2. When you need to push to defend UX work vs. compromise
3. When you need to ask for resources/help
4. When you have enough to get started and clarify with later iterations

Additionally, there are plenty of other soft skills like:

1. How to work with different teams and personalities
2. How to balance the wants and needs of different users, stakeholders, developers and maybe even external vendors
3. How to properly deliver in a way that highlights the crux of the work you did without getting lost in details