Example 2 (2004 National High School Competition Question) As shown in Figure $7-3$, the axis section of a cone with vertex $P$ is an isosceles right triangle, $A$ is a point on the circumference of the base circle, $B$ is a point inside the base circle, $O$ is the center of the base circle, $A B \perp O B$, with the foot of the perpendicular at $B, O H \perp P B$, with the foot of the perpendicular at $H$, and $P A=4, C$ is the midpoint of $P A$. When the volume of the tetrahedron $O-H P C$ is maximized, the length of $O B$ is ( ).
A. $\frac{\sqrt{5}}{3}$
B. $\frac{2 \sqrt{5}}{3}$
C. $\frac{\sqrt{6}}{3}$
D. $\frac{2 \sqrt{6}}{3}$