What's the deeper reason behind the distinction of coordination and subordinatio...

I often see grammar books define a subordinate clause as “a clause that cannot stand alone.” However, this definition feels vague to me.

What exactly does “cannot stand alone” mean here? Does it mean semantically incomplete, pragmatically odd, or grammatically dependent?

For example, in a sentence like:

Since I was ill, I did not come.

many teachers say that since I was ill is a subordinate clause because it “sounds incomplete” or “cannot stand alone.” But that explanation does not seem convincing to me, because almost any clause can sound odd when separated from its context.

For instance, in:

He worked hard, but he failed.

the clause: "but he failed"

also sounds strange if read by itself, yet we do not call it a subordinate clause.

So I do not fully understand what distinguishes subordinate clauses from coordinated clauses.

Another common explanation is that subordinate clauses are introduced by “subordinating conjunctions” such as since, although, unless, while, when, etc. But this also feels somewhat circular to me:

1. We say a clause is subordinate because it begins with a subordinating conjunction.
2. But then how do we decide that certain conjunctions are subordinating while others (like and, but, or or) are coordinating?

What is the deeper grammatical difference between coordination and subordination?

I checked the answer to this on this website but they all said "you'd have to read the detailed explanation in The Cambridge grammar of the English language" but that book has a lot of jargon that I cannot understand also I don't even know exactly which section I have to read to get the answer.

I would especially appreciate an explanation in simple structural terms rather than only traditional textbook definitions since I am not someone who has deep knowledge of the English language and grammar.

In the simplest terms: subordinate clauses cannot change places with the superordinate ones while keeping the meaning roughly the same, while coordinate clauses can be swapped around without affecting the logic of the sentence.

• He worked hard, but he failed.

means basically the same as

• He failed, but he worked hard.

Whereas

• Since I was ill, I did not come

does not mean

• Since I did not come, I was ill.

Of course there's a little more subtlety to it than that, but then we'd have to get into further examples, and making an exhaustive list of all types of subordinate clauses and delving into the details of the structures that allow them is best left to reference grammars like the ones in the OP. In any case, the general principle above holds quite well.