This question is a response to the answer given in this SE post, concerning the use of the Vitali covering lemma to prove the Hardy-Littlewood maximal inequality in general dimension. For reference, the covering lemma is
Lemma (Vitali). Let $B_1,\ldots,B_n$ be a finite collection of open balls in $\mathbb{R}^d$ (not necessarily disjoint). Then there exists a subcollection $B_1',\ldots,B_m'$ of disjoint balls in this collection, such that $$ \bigcup_{i=1}^n B_i \subseteq \bigcup_{j=1}^m 3B'_j, $$ where $cB(x,r) := B(x,cr)$ for $c>0$. (In fact, each $B_i$ is contained fully in some $3B'_j$.)
I'm interested in the following claim made in the linked post:
Suppose $K \subseteq \mathbb{R}^d$ is a compact set, and for every $x \in K$, we are given an open ball $B(x,r_x)$ that is centered at $x$ and of radius $r_x$. Assume that $$ R := \sup_{x \in K} r_x <\infty. $$ Let $\mathcal{B}$ be this collection of balls, i.e. $$ \mathcal{B} := \{B(x,r_x) : x \in K\}. $$ Then given any $\epsilon > 0$, there exists a finite subcollection $\mathcal{C}$ of balls from $\mathcal{B}$, such that the balls in $\mathcal{C}$ are pairwise disjoint, and such that the (concentric) dilates of balls in $\mathcal{C}$ by $(2 + \epsilon)$ covers $K$.
This result can be used to improve the proportionality constant in the Hardy-Littlewood maximal inequality from $3^d$ to $2^d$, but my question concerns the proof of this result. Why do we need to assume the sup of the radii of the $B(x,r_x)$ is finite? I came up with a proof that doesn't seem to need this fact at all (and indeed, it seems to me that if that hypothesis were necessary, we would be unable to apply the result to improving the Hardy-Littlewood maximal inequality as in the linked post.)
My argument is as follows: The collection $\{B(x,\epsilon r_x) : x \in K\}$ covers $K$, and so by compactness of $K$ we may extract a finite subcover $\{B(x_i,\epsilon r_i) : 1 \leq i \leq n\}$. Now apply the construction from the proof of the Vitali-type covering lemma to the collection $\{B(x_i,r_i) : 1\leq i \leq n\}$ (notice the difference between this collection and the previous) to yield a subcollection of balls $\mathcal{C} := \{B(x_{i_j}, r_{i_j}) : 1 \leq j \leq m\}$.
We shall show that $\mathcal{C}$ is a suitable subcollection for the lemma. Let $x \in K$. Then since $\{B(x_i,\epsilon r_i) : 1 \leq i \leq n\}$ covers $K$, we have $|x - x_i| < \epsilon r_i$ for some $i \in \{1,\ldots,n\}$. Recall from the proof of the Vitali-type covering lemma that $B(x_i,r_i)$ must intersect some $B(x_{i_j},r_{i_j})$ such that $r_{i_j} \geq r_i$. Thus $|x_i - x_{i_j}| < r_i + r_{i_j} \leq 2 r_{i_j}$ by the triangle inequality. Again by the triangle inequality, we obtain $|x - x_{i_j}| < \epsilon r_i + 2r_{i_j} \leq \epsilon r_{i_j} + 2r_{i_j} = (2 + \epsilon)r_{i_j}$, and thus $x \in (2 + \epsilon)B(x_{i_j},r_{i_j})$. The claim follows.
Is that proof correct? Does it rely on the radii of the original balls being uniformly bounded?
[EDIT: Found a problem with my original argument and replaced it with a different argument.]