For a set $C\subset \mathbb R^2$, define its visibility from a point $x$ as $vis_C(x)=\{\varphi\in \mathbb S^1\mid \exists t>0~~x+t*\varphi\in C\}$, where $\mathbb S^1$ denotes the unit circle. Say that $C$ looks $m$-big from $x$ if $\mu(vis_C(x))/2\pi=m$, where $\mu$ is the (outer) Lebesgue measure on $\mathbb S^1$.

If $C$ is at most $0.99$-big from every $x$, then is $\mu(C)=0$?

Here again $\mu$ can be the Lebesgue measure, but I'm interested in any similar results as well. My motivation is to solve this problem. The question also seems somewhat related to this one.

Note that the converse is false, as if $C$ is a line, then it looks $1/2$-big from every $x\in \mathbb R^2\setminus C$, and if we take the union of countably many horizontal lines, we can construct a set that looks $1$-big.