FALSE: the Vitali covering theorem holds for arbitrary covers

Consider the cover of $[0,1]$ by all intervals $[0,t]$ and $[t,1]$ with $0<t<1$. It covers every point of $[0,1]$, but it is not fine: for an interior point $x$, every interval in the cover that contains $x$ has length at least $\min \{x,1-x\}$.

Any two left intervals intersect, and any two right intervals intersect, so a disjoint subfamily contains at most one interval of each type. If it contains only one interval, it obviously misses points of $[0,1]$. If it contains one left interval $[0,s]$ and one right interval $[t,1]$, disjointness forces $s<t$, so the open gap $(s,t)$ is uncovered. Thus no disjoint subfamily covers $[0,1]$ up to a null remainder, and the statement is false.