Concentration of norm of projection onto a subspace

Let $x$ be a random vector uniformly distributed on the unit sphere $\mathbb{S}^{n-1}$. Let $V$ be a linear subspace of dimension $k$ and let $P_V(x)$ be the orthogonal projection of $x$ onto $V$. I have seen quoted in the literature that \begin{align} \mathbb{P}[|\left\| P_V(x)\right\|_2 - \sqrt{k/n} | \le \epsilon] \ge 1 -2\exp(-n\epsilon^2/2). \, \, \, \, \, \, \, (1) \end{align} However, i can still not find a concrete proof. What i do understand is that for a $1$-Lipschitz function $f:\mathbb{S}^{n-1} \rightarrow \mathbb{R}$ such as $x \mapsto |\left\| P_V(x)\right\|_2$, we have that \begin{align} \mathbb{P}[|f - M_f | \le \epsilon] \ge 1 -2\exp(-n\epsilon^2/2), \, \, \, \, \, \, \, (2) \end{align} where $M_f$ is the median of $f$. (2) mostly follows from the isoperimetric inequality on the sphere. The issue though with (1) is that $\sqrt{k/n}$ does not seem to be the median of $x \mapsto |\left\| P_V(x)\right\|_2$. Is anyone able to provide a clean argument for (1) or a self-contained reference in the literature? Many thanks.

One feature of concentration of measure is that once you know concentration around some of the natural "averages" (mean, median, a given quantile, a given $L^p$ norm), you can derive formally that concentration holds around all of them.
In your case $\sqrt{k/n}$ is the $L^2$-average of the function $f : x \mapsto \|P_Vx\|_2$.
This discussed in Chapter 5.2 in G.Aubrun and S.J.Szarek, Alice and Bob meet Banach:The Interface of Asymptotic Geometric Analysis and Quantum Information Theory), for the case of $L^2$ average see Exercise 5.46. This possibly won't give the constants you claim in (1).
All k dimensional projections are the same, so you might as well look at the first K co-ordinates. The norm squared of the projection can be represented as $\frac {X_1^2 + ... + X_k^2}{ X_1^2 + ... + X_n^2}$ where the $X_i$ are i.i.d. N(0,1). I think this actually has a beta distribution. I haven't tried to get your numbers out of this but this framework seems simple.