I) Let us for simplicity put the physical constants $\hbar=1=m$ to one. OP is considering the usual transcription $u(r)\equiv rR(r)$ of the 3D radial TISE into a 1D TISE,
$$ - \frac{1}{2} u^{\prime\prime}(r)+U_{\ell}(r)u(r) ~=~E u(r),\tag{A}$$
where the total potential energy
$$ U_{\ell}(r)~:=~ U(r) + \frac{C_{\ell}}{2r^2} \tag{49.8b}$$
is a sum of a central potential energy $U(r)$ and a centrifugal potential energy
$\frac{C_{\ell}}{2r^2}$. Here and below the equation numbers refer to Ref. 1. The constant
$$ C_{\ell}~:=~\ell (\ell +1)~=~\left(\ell+\frac{1}{2}\right)^2 -\frac{1}{4} \tag{B}$$
in eq. (49.8b) is the eigenvalue of the $\hat{L}^2$ operator.
II) We are investigating a bound state where the angular momentum $\ell>0$ is non-zero.
We are interested in the situation where the centrifugal potential energy numerically completely dominates [and the potential $U(r)$ can be ignored] in a neighborhood $[0,r_0+\epsilon[$ of the classically forbidden interval $[0,r_0[$, where $r_0$ denotes the inner radial turning point. In other words,
$$ E~\approx~\frac{C_{\ell}}{2r_0^2}. \tag{C}$$
We also want the semiclassical WKB approximation to be valid in the interval $[0,r_0[$ (away from the turning point). The semiclassical condition
$$ |\lambda^{\prime}(r)|~\ll~ 1 \tag{46.6} $$
implies that
$$ \ell ~\gg~ 1.\tag{D}$$
III) The (absolute value of the) momentum is
$$\begin{align} p(r)~:=~&\sqrt{2\left|E - U_{\ell}(r)\right|}\cr
~\approx~&\sqrt{C_{\ell}\left|r^{ - 2} - r_0^{-2}\right|} \quad\text{for}\quad r\in[0,r_0+\epsilon[, \end{align} \tag{46.5}$$
where
$$ \sqrt{C_{\ell}} ~\stackrel{(B)}{=}~\ell+\frac{1}{2}-\frac{1}{8\ell}+{\cal O}(\ell^{-2}). \tag{E} $$
The semiclassical connection formulas yield
$$ u(r)~\approx~\frac{c}{\sqrt{p(r)}}\exp\left[ \int_{r}^{r_0} \! dr^{\prime}~p(r^{\prime})\right]
\quad\text{for}\quad r~<~ r_0, \tag{F} $$
$$ \begin{align} u(r)~\approx~&\frac{c}{\sqrt{p(r)}}\cos\left[ \int_{r_0}^{r} \! dr^{\prime}~p(r^{\prime})-\frac{\pi}{4}\right]\cr
~=~&\frac{c}{\sqrt{p(r)}}\sin\left[ \int_{r_0}^{r} \! dr^{\prime}~p(r^{\prime})+\frac{\pi}{4}\right]
\quad\text{for}\quad r~>~ r_0, \end{align} \tag{48.1} $$
where $c\in\mathbb{C}$.
IV) The semiclassical approximation (F) behaves as
$$ u(r)~\propto~ r^{\sqrt{C_{\ell}}+\frac{1}{2}}\quad\text{for}\quad r~\to~ 0^{+}, \tag{G}$$
while the well-known exact behavior is
$$ u(r)~\propto~ r^{\ell+1}\quad\text{for}\quad r~\to~ 0^{+}. \tag{32.15}$$
Hence, the semiclassical approximation would have the correct behavior at the origin $r=0$ if we replace (E) with $\ell+\frac{1}{2}$.
V) Alternatively, in the free case $U(r)=0$, the semiclassical approximation (48.1) behaves as$^1$
$$ u(r)~\propto~\sin\left[\sqrt{C_{\ell}}\left(\frac{r}{r_0}-\frac{\pi}{2}\right)+\frac{\pi}{4}\right] \quad\text{for}\quad r~\to~\infty, \tag{H} $$
while the well-known exact behavior is
$$ u(r)~\propto~\sin\left[\sqrt{C_{\ell}}\frac{r}{r_0}-\ell\frac{\pi}{2}\right] \quad\text{for}\quad r~\to~\infty. \tag{33.12} $$
This again suggests to replace (E) with $\ell+\frac{1}{2}$.
References:
- L.D. Landau & E.M. Lifshitz, QM, Vol. 3, 3rd ed, 1981; $\S49$.
$^1$ Momentum integral becomes
$$\begin{align} \frac{1}{\sqrt{C_{\ell}}} \int_{r_0}^{r} \! dr^{\prime}~p(r^{\prime})
~=~&\int_1^{\frac{r}{r_0}} \! \frac{dx}{x}\sqrt{x^2-1}\cr
~=~&\left[ \sqrt{x^2-1}+\arctan\frac{1}{\sqrt{x^2-1}} \right]_{x=1}^{x=\frac{r}{r_0}}\cr
~\approx~&\frac{r}{r_0}-\frac{\pi}{2}\quad\text{for}\quad r~\to~\infty. \end{align} \tag{I} $$
