I have a collection of $n$ unit-radius disks in $\mathbb{R}^3$, whose centers are random within a sphere of radius $R>1$, and which are each oriented randomly. I'd like to find a line $L$ that pierces as many disks as possible: $L$ "spears" the "hoops," the circle boundaries of the disks.

This question arose thinking about accidental line-of-sight alignment of galaxies.

^{ $n=100$ disks centers within $R=10$ of origin (red). $L$ pierces $5$ disks. }

. What is growth rate of the expected largest number of piercings by a line $L$, growth with respect to $n$ as $R$ remains fixed?Q1

I expect it grows more slowly than linearly with $n$.
** Update**. @YCor shows in a comment that, in fact, it grows at least linearly.

I'd like to find a max-piercing $L$ without using Plücker coordinates and intersecting regions on the Grassmannian manifold, which I expect to be a non-trivial implementation challenge.

. Is there an "easy" way to find a max-piercing $L$, even if computationally inefficient?Q2