2-body problem, conservative central force, spherical symmetry, conversation of angular momentum, motion in a plane

I am having trouble unpacking the following argument from page 72 of Goldstein’s classical mechanics book.

We now restrict ourselves to conservative central forces, where the potential is V(r), a function of r only, so that the force is always along r. By the results of the preceding section, we need only consider the problem of a single particle of reduced mass m moving about a fixed center of force, which will be taken as the origin of the coordinate system. Since potential energy involves only the radial distance, the problem has spherical symmetry; i.e., any rotation, about any fixed axis, can have no effect on the solution. Hence, an angle coordinate representing rotation about a fixed axis must be cyclic. These symmetry properties result in a considerable simplification in the problem.
Since the problem is spherically symmetric, the total angular momentum vector,
$$L = r times p$$,
is conserved. It, therefore, follows that r is always perpendicular to the fixed direction of L in space. This can be true only if r always lies in a plane whose normal is parallel to L. While this reasoning breaks down if L is zero, the motion, in that case, must be along a straight line going through the centre of force, for L = 0 requires r to be parallel to $dot{r}$, which can be satisfied only in straight-line motion.* Thus, central force motion is always motion in a plane.

Firstly, how does it follow from assuming that the potential in the problem is only dependent on the radial distance between the two bodies, that the system is spherically symmetric? I don’t have a problem understanding that the problem IS spherically symmetric, but isn’t it more? It has translational symmetry too, no? Is the author only mentioning spherical as we’re concerned with the conservation of ang. mom. at the moment?

Secondly, by a combination of this section in Wikipedia and knowledge of Noether’s theorem, I can work out that spherical sym -> L=conserved -> r and L are perpendicular. What I’m struggling with is how $dot{r}$ is ALSO perpendicular to L (and moreover parallel to r) to state that 2 body motion occurs in a plane.

The footnote Goldstein includes is

$dot{textbf{r}}= dot{r}textbf{e}_r + rdot{theta} textbf{e}_theta$, hence $textbf{r}times dot{textbf{r}} =0$ requires $dot{theta}=0$

but I’m unsure how this relates back to $L=0$.

The above mentioned Wiki article is more clear to me, but I’m still struggling with the jump from

$$L=r times p = rtimes mu dot{r} text{ is constant}$$

to $dot{r}$ must be perpendicular to L as well.

Any help with some intuition is appreciated!