After struggling with handstands for, well, 7-8 years in all, I finally noticed a small point mentioned in a video tutorial for some other skill. It demonstrated a handstand as a pre-requisite and the tip is simply this:
Kick up as slowly as you can, and bring the leading leg up to the balance long before the second one. Then raise your second leg slowly to meet the first, feeling your balance the whole way.
Bringing the second leg up feels very slow, but it's really only one or two seconds. But you consciously bring it up, under control, until it's just right.
Apparently many people do this intuitively, but the idea apparently completely passed me by for quite a while. I used to just kick up both legs very quickly (one after the other), and only chanced upon the balance point maybe one in 10-20 times. Now I suddenly can get up around half the time, and am holding 10-30s handstands very easily - finally often enough and for long enough to start trying variations or straightening out my body shape.
I hope this helps some of you!