Here's a proof that you can't place 5 ligands equidistant from each other and from the central atom in a nonplanar molecule:
- Place two points anywhere you like on the surface of a sphere around the central atom.
They will define a great circle (an "equator") that cuts the sphere in half.
- You have 3 points left. You can't place them all off the "equator" without putting two in one hemisphere and 1 in the other. You cannot make all the points equidistant. The only way you can balance the number of points in each hemisphere is to put one in one hemisphere, one in the other, and one on the "equator" with the first two you placed. But the distance between the equatorial points will not be the same as all of the distances to points in the hemisphere.
Why is it trigonal pyramidal, though? The best we can do is make subsets of the ligands equidistant. Place three equidistant points on the equator, and two at the "poles". Imagine a linear shape (the axial positions) passing at right angles through a trigonal planar shape (the equatorial positions). You have a trigonal bipyramidal shape.
The dsp3 hybridization is a futile attempt to explain 5 orbitals around the central atom by mixing the central atom's orbitals alone. In fact back in the early 1990's it was shown that this is a poor explanation (source: E. Magnusson, Hypercoordinate molecules of second-row elements: d functions or d orbitals? J. Am. Chem. Soc. 1990, 112 (22), 7940–7951. DOI: 10.1021/ja00178a014.)