Yes?

If I use 3 shares and require all three to recover, then I think I could let the 3 points define a plane. Then the plane would intersect the axis at a unique point.

If I want 4 or more shares, I can make them by just generating more points on that same plane. Then I think any three of those should recover the secret. So it seems like that would all work about the same as with the polynomials? Then I can go to still more dimensions when I want to require more shares to reveal the secret.

Probably, but the nice thing about polynomials is the secret is the value when x is 0. What is the point on the hyper geometry where I look up the secret?

Two points make a line in any number of dimensions.