The green insect is an aphid. But you're right, that's not what is causing the damage. Where are you located? That would have an influence on what pests you might get.

TOronto. Both plants have been indoors, by the window, in the kitchen, just above the sink.

To be honest, when I googled imaged downy mildew basil, I was certain that's what I had. And also sooty mold, which can come from the honeydew of aphids. But then on the other thread people kept saying it's not mildew, so I don't know.

Forgot to mention stunted growth as a symptom. They've been really slow to grow and seem to have hit a point beyond which they haven't grown.

It's hard to tell from the photos, but if you thought it was downy mildew on the basil then I will say that grey mould (Botrytis) has some signs that look quite similar to basil downy mildew. And that would spread to the thyme, whereas downy mildew would not.

Again it's hard to tell but it looks a little bit like it in the first photo, where there is some dieback and greying of the thyme leaves.

It doesn't explain the black spots though. I honestly think those are more likely to be either insects or something cultural, rather than disease. There is some clear evidence of insect damage - in the third basil photo and the third thyme photo.