Ok, so I have a new hypothesis about why Bridget likes to gesture with her forelegs. She never seems to want anything in particular - she enjoys it when I pull on her leg, or scratch it, or hold it up, but none of these appear to be her goal.
Two things got me thinking. One was that wee Islay of Ardtornish did exactly the same thing. And the other was an incident that happened a couple of days ago. I was out in the field, and Bridget came up. I was happy to see her and gave her a big hug, followed by much vigorous rubbing of her head and neck. (So gratifying to have a horse who enjoys that kind of thing!) She responded by going into overdrive with her leg gestures - throwing her foreleg at me repeatedly with great gusto, matching the enthusiasm with which I'd greeted her.
So then it occurred to me: they're doing this in imitation of a human using its hands.
That's my theory, and I'm sticking to it.