Dealing with unsupportive committee / adviser after completion of Ph.D.?

We don't know if your advisor didn't respond out of passive-aggressiveness, or through a slip-up. I recommend you assume the latter and try again.

You can also try phoning or visiting during office hours or some time you think he's likely to be in. And/or leaving a message with the department secretary.

If you get clearer evidence he is intentionally ignoring your request, you can then approach the graduate program director and explain your problem. S/he should either arrange for a departmental letter of reference (as might be done if the advisor is not available to write a letter), or give your advisor a push.

If that doesn't get you anywhere, you can try going up the administrative chain of command.

There is an alternate approach, and I recommend you pursue it concurrently: take or audit a class from someone you think would be in a position to write a strong letter. You might do this at your home institution or somewhere else.

Aside from all that -- I hope you have lined up your outside reader for a letter.

I'm sorry that you had such a negative advising experience, and in particular that faculty at your institution were unsupportive (and worse) about your health problems.

As you say, you finished your PhD a year and a half ago. The good news is that you're well out of that highly unpleasant environment, and I hope you're experiencing the benefits of that. The bad news? Because of what you're asking, I really wonder what you've been doing professionally for the last year and a half.

I don't necessarily expect my references to be glowing. But am I right in thinking that I am at least owed a response to a request for a reference?

Yes, of course. I'm sorry that you even have to ask.

And, without that reference, am I correct in thinking that I have basically no chance of teaching again?

No, that's too strong. It would most certainly be better to have references from within your department, but that's not always possible. Sometimes the one reference from within your department would have come from your thesis divisor, but s/he died. Sometimes good people are not viewed well by their PhD-granting departments. It's not good, but it's not the end either.

I do have references from faculty in other departments where I worked as an adjunct/sessional lecturer[.]

Great, do that.

[B]ut they can't really speak to my research skills[.]

Well, good news: there are a lot of academic teaching jobs for which your research skills are largely to completely irrelevant.

[A]nd my feeling is that the lack of anything from my own department makes me professionally untouchable.

Can you get a letter from someone in your department, even if they are only superficially familiar with your research? That doesn't make a lot of sense logically, but psychologically it allows people to tick a box, so it could be helpful. If you can do that and get a strong recommendation from your outside reader, then I think your problems are pretty close to being solved.

Finally: please don't wait a year and a half wondering whether someone will reply to an email you sent. You should have checked in about ten times by now. Again, I am really sorry for your poor treatment: one of the worst aspects of being treated poorly is that it makes you expect to be treated poorly. That sucks.