This is pretty subtle!
Are you familiar with the concept of presuppositions? For example, a sentence like "My son is in high school" tacitly assumes that the speaker has a son. This tacit assumption is called a "presupposition".
One major use of the word if is in so-called "counterfactual" conditionals, which presuppose that their hypothesis is false. For example, a sentence like "If Alice were here, she would know what to do" tacitly assumes that Alice is not present.
But there's a subtlety here. Sometimes the hypothesis is false, and the speaker knows that the hypothesis is false, but instead of presupposing this, the speaker rhetorically entertains the possibility that it might be true, in order to demonstrate that it actually isn't. For example, in a sentence like "Bob would have eaten more of his dinner if he'd been very hungry", the speaker doesn't have any direct information about how hungry Bob was, but is arguing that Bob wasn't very hungry by saying that if he had been, then he would have eaten more of his dinner.
Consider a sentence like this one:
He would be here if he could.
Without context, we can't tell whether it's presupposing that its hypothesis ("he can be here") is false, or merely arguing that it is.
Here's an example where it is presupposing:
I just got off the phone with Bob. His flight has been canceled, so he won't be able to make it in time. He said he's very sorry. He would be here if he could.
and here's an example where it is not:
I haven't been able to get hold of Bob, but I know that he was really looking forward to this meeting, so I'm sure that there must be a good reason that he's not here. He would be here if he could.
So, with that background, I can now answer your question!
- We can use unless if we're arguing that something is true by entertaining the possibility that it's false.
- But we can't use unless if we're simply presupposing that something is true.
So a sentence like this one:
He would be here unless he couldn't for some reason.
doesn't have the subtle ambiguity that "He would be here if he could" does.
In the BBC World Service post that you looked at, this is what Roger Woodham is getting at with the examples that you quote:
- If he didn't take everything so seriously, he would be much easier to work with.
- If he weren't so bad-tempered, I would help him to get the work done.
- If you hadn't driven so recklessly, you wouldn't have had this accident.
- If you hadn't had that last glass of wine, this would never have happened.
We can see that these are making presuppositions because of how they use so and that to refer to specific quantities that are (apparently) already known to both speaker and audience.
In the Education First page that you looked at, all of the example sentences with unless are fine, but some of the explanatory text is not. In particular:
- It says that "Unless is used instead of if...not in conditional sentences of all types", but that's not really true, because if is used in cases where the hypothesis is presupposed to be false, where unless is not used that way. (The problem here is that the explanatory text classifies conditional sentences in a simplistic way — only three "types" — which doesn't capture all of the different possible subtleties.)
- It describes the various sentences with unless as being "equivalents" of corresponding sentences with if; but many of the sentences with if are ambiguous in the same way as "He would be here if he could", which means that the corresponding sentences with unless aren't fully equivalent, because they aren't ambiguous in that way.
Incidentally, "if" can also be used in cases where the hypothesis is presupposed to be true, and "unless" can't be used in those cases, either!
For example, if Carol is supposed to be stationed at the entrance to let people in, but then you run into her somewhere else, then you might say "If you're not at the entrance, then who's letting people in?", but you wouldn't say *"Unless you're at the entrance, who's letting people in?"
